On concentric fractal spheres and spiral shells
Efstathios Konstantinos Chrontsios Garitsis





Biermann-battery driven magnetized collisionless shock precursors in laser produced plasmas
    C. K. Li
    September 9, 2024
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§ ABSTRACT 
We investigate dimension theoretic properties of concentric topological spheres, which are fractal sets emerging both in pure and applied mathematics. We calculate the box dimension and Assouad spectrum of such collections, and use them to prove that fractal spheres cannot be shrunk into a point at a polynomial rate. We also apply these dimension estimates to quasiconformally classify certain spiral shells, a generalization of planar spirals in higher dimensions. This classification also provides a bi-Hölder map between shells and constitutes an addition to a general programme of research proposed by J. Fraser in <cit.>.

Keywords. Fractal, dynamical systems, spiral, Assouad spectrum, quasiconformal mappings.





§ INTRODUCTION




Concentric objects occur in various areas of pure and applied mathematics. In complex dynamics, the Fatou components of entire transcendental maps are typically of fractal boundary and converge to infinity (see for instance <cit.>). Therefore, under a conformal inversion that keeps most geometric properties intact, the components of the map can be seen as fractals “centered" at 0. In the context of harmonic analysis, concentric Euclidean spheres are closely associated to weights of measures that lie in the important Muckenhoupt class <cit.>. In fact, polynomially-concentric sphere collections, i.e., unions of spheres S_n centered at 0 with radii equal to n^-p, for some p>0, provide one of the very few examples of bounded sets whose distance function lies in the Muckenhoupt class, for certain exponents. 

A set S⊂^d is called a topological sphere, if there is a homeomorphism f:^d→^d which maps the unit Euclidean sphere centered at 0, denoted by S(0,1)⊂^d, onto S. A topological sphere can potentially be smooth, or extremely “rough". 
A quantitative classification between the two cases is a difficult problem of many interpretations. As a result, it has received attention within numerous fields, such as topology, analysis on metric spaces and geometric group theory (see <cit.> for a detailed survey on the topic). Despite such difficulties on the theoretical front, concentric topological spheres provide examples of concentric fractals that emerge in various applied areas, such as mathematical physics <cit.>, <cit.>, machine learning <cit.>, and dynamic scaling of mathematical models <cit.>. 





It might seem natural that very rough fractals, which are disjoint and centered at a fixed point, would have to decrease a lot in size while they approach the center. More specifically, in concentric  topological sphere collections, where the spheres are allowed to be fractals of high fractal dimension, they need to converge to their center significantly fast in order to avoid overlaps. Hence, for any concentric collection of topological spheres, there are  certain restrictions on either their dimension, or on how fast they accumulate around their center. One of our main results is addressing this phenomenon for fractal spheres, by studying certain dimension notions for the whole collection. For what follows, we fix an integer d≥ 2, and we denote by B(0,1)⊂^d the open unit ball in ^d centered at 0. We formalize the notion of concentric sphere collections in the following definition.






    Let S_0⊂^d be a topological sphere with S_0= f(S(0,1)), for some homeomorphism f:^d →^d. For x_0∈ f(B(0,1)) and a sequence of similarities {g_n}_n∈, set S_n:=g_n(S_0). We say that the union S:=⋃_n∈_0 S_n is a concentric sphere collection centered at x_0 and generated by S_0, if there are c_1, c_2∈ (0,1], n_0∈, and a strictly decreasing sequence {a_n}_n∈ converging to 0 such that:
    
        
  * d_H(S_n,{x_0})=c_1 a_n, and
        
  * (S_n,S_n+1)=c_2 (a_n-a_n+1),
     for all n≥ n_0.
    Moreover, if there is p>0 so that  a_n=n^-p, for all n≥ n_0, then we say that S is a p-(polynomially) concentric sphere collection centered at x_0 and generated by S_0, and we denote it by S_p=S_p(x_0,S_0).


An example of a planar concentric sphere collection is depicted in Figure <ref>, where S_0 is the Koch snowflake (see <cit.>). Note that in Definition <ref>, (S_n,S_n+1) denotes the usual Euclidean distance between the sets S_n,S_n+1, and d_H denotes the Hausdorff distance (see Section <ref> for the definition). The latter is a natural way to talk about convergence of a collection of sets into a point (see for instance <cit.>, <cit.>).
Therefore, one could perceive Definition <ref>  as the process of “shrinking" the topological sphere S_0 into a point x_0, using contracting similarities g_n of ratio approximately a_n.
Such problems have attracted a lot of interest from various areas of mathematics, and they have been extensively studied both from a deterministic <cit.> and a probabilistic <cit.> point of view.
One of the main contributions of this paper is to emphasize that dimension-theoretic techniques can be crucial to the understanding of this phenomenon. Namely, we adopt such techniques in order to prove that a topological sphere S_0 of non-trivial box dimension _B S_0 (see Section <ref> for the definition) cannot give rise to polynomially concentric sphere collections of  any degree.






    Let S_0⊂^d be a topological sphere with _B S_0∈ (d-1,d]. For any p>0, there is no p-concentric sphere collection generated by S_0.


It should be noted that it is always possible to shrink a topological sphere S_0 into a point through a sequence of similarities as described in Definition <ref>, as long as the sequence a_n converges fast enough to 0. See Section <ref> for a detailed discussion. Theorem <ref> establishes that sequences of the form n^-p (or of slower rate of convergence to 0) are simply not fast enough to shrink a topological sphere of non-trivial box dimension. Hence, all concentric fractal spheres encountered in the aforementioned applications <cit.> either “smoothen" out as they approach the common center, or they shrink at an exponential rate. 
On the other hand, there are still various examples of topological spheres in ^d of box dimension equal to d-1 that generate polynomially concentric collections. For instance, the boundary of any convex set homeomorphic to B(0,1)⊂^d can be polynomially shrunk into its center of mass. 


The proof of Theorem <ref> is achieved by assuming towards contradiction that for d_0>d-1 and p>0, a p-concentric sphere collection S_p(x_0,S_0)⊂^d exists. We then use dimension theoretic tools, and specifically the Assouad spectrum notion {_A^θ S_p: θ∈ (0,1)} introduced by Fraser-Yu <cit.> (see Section <ref> for the definition), in order to show that such a collection would be of dimension larger than d,  surpassing the dimension of the ambient space ^d and leading to a contradiction. A partial version of Theorem <ref> can be proved by calculating the box dimension of such a collection, which  results in a contradiction only for certain values of p. Hence, the Assouad spectrum is  essential in achieving the result in full generality with this approach. Theorem <ref> is the first application of the Assouad spectrum in this context of “rigidity" of fractal collections.
Moreover, the property of the Assouad spectrum to trace “finer" geometric data is also crucial for  “classification" problems under families of mappings. We establish Assouad spectrum estimates for concentric sphere collections (see Section <ref>), which we apply to classify certain objects, up to classes of homeomorphisms. 







An instance of a planar set resembling a collection of concentric circles is the spiral of the form

    (ϕ):={ϕ(t) e^it:   t>1 },
 where ϕ:[1,∞)→(0,∞) is a continuous, strictly decreasing function, with lim_t→∞ϕ(t)=0. Indeed,  such spirals share many geometric properties with  concentric collections of circles of the form ∪_n∈ S(0,ϕ(n)), while they hold a prominent role in fluid turbulence <cit.>, dynamical systems <cit.>, and even certain types of models in mathematical biology <cit.>. Moreover, they provide examples of “non-intuitive" fractal behavior (see <cit.>), while they have also been extensively studied due to their unexpected analytic properties. For instance, Katznelson-Nag-Sullivan <cit.> demonstrated the dual nature of such spirals, lying in-between smoothness and “roughness", as well as their connection to certain Riemann mapping questions. The relation between the rate of convergence of ϕ(t) for t→∞ and the existence of Lipschitz and Hölder parametrizations has also been extensively studied by the aforementioned authors in <cit.>, by Fish-Paunescu in <cit.>, and by Fraser in <cit.>. In particular, Fraser in the latter paper focuses on spirals that resemble polynomially concentric circles, i.e., spirals where ϕ(t)=t^-p, for p>0, and emphasizes further the “fractal" nature of these objects, due to the peculiar behavior they exhibit in relation to certain dimension notions.


A natural higher dimensional generalization of spirals is the notion of spiral shells in ^3. They exhibit similar behavior to spirals, which has attracted the interest of researchers in physics <cit.> and mathematical biology <cit.>. For p>0, we consider polynomial (spiral) shells of the form

    _p:= { (u^-pcos u sin v, u^-psin u sin v, u^-pcos v)∈^3:   u,  v∈ [1,∞) }⊂^3.
 The resemblance to the polynomial spirals mentioned above is perhaps already evident from the definition. In addition, spiral shells have a very similar geometric structure to p-polynomially concentric spheres in ^3. See  Figure <ref>.




One significant restriction in the analytic and geometric study of these objects is the lack of planar tools in ^3. For instance, Liouville's theorem states that all conformal maps are Möbius transformations in ^3 (see for instance <cit.>). This was one of the motivations for the definition of quasiconformal maps in ^d, d≥ 2 (see Section <ref> for formal definition). More specifically, while conformal maps transform infinitesimal balls into infinitesimal balls, quasiconformal maps transform them into infinitesimal ellipsoids of globally bounded eccentricity. The bound on said eccentricity is called dilatation. If the dilatation of a quasiconformal map f is at most some uniform constant K≥ 1, we say that f is a K-quasiconformal map (see Section <ref>). The theory of quasiconformal maps has a rich history, which still continues to develop actively, especially due to its many connections and applications within various fields of mathematics. We refer to the books by Lehto and Virtanen <cit.>, Väisälä <cit.>, and Heinonen <cit.> for thorough expositions on these maps and their importance in , ^d and metric spaces, respectively.

Distinguishing sets up to quasiconformal maps, particularly at the level of the dilatation, is in general a hard problem.
For instance, determining under what assumptions a topological sphere is a quasiconformal image of the usual unit sphere S(0,1) is a long-standing open problem, with many implications in geometric group theory and topology (see <cit.> for a survey on this topic and connections to Cannon's conjecture, see also <cit.>). For planar topological spheres, i.e., simple closed curves homeomorphic to the unit circle, this classification problem was solved by Ahlfors in <cit.>, and their dimension-theoretic properties were fully determined by Smirnov in <cit.>. In higher dimensions only partial progress has been made; see <cit.> and the references therein. 


Given the ties of spiral shells to several applications, as well as their relation to concentric topological spheres, it is natural to try and classify them under quasiconformal maps. Indeed, this is the outcome of our second main result. By applying the Assouad spectrum estimates we establish for concentric spheres (see Theorem <ref>), along with the dimension distortion properties of quasiconformal maps, proved by Tyson and the author in <cit.>, we achieve the desired classification of polynomial shells.


    For p≥ q>0, let _p and _q be two polynomial spiral shells. There is a K-quasiconformal mapping F:^d→^d with F(S_p)=S_q if, and only if, K≥ p/q.
    




Note that quasiconformal maps are also locally Hölder continuous, with locally Hölder continuous inverse (see Section <ref> for definitions). Therefore, in the proof of Theorem <ref>, the quasiconformal map we construct, mapping _p onto _q, is a bi-Hölder map between polynomial spiral shells, with the Hölder exponents quantitatively depending on the contracting ratios of the shells p, q. This contributes to a general direction proposed by J. Fraser in <cit.>, on bi-Hölder classification of sets in ^d. We discuss this connection further in Section <ref>.

This paper is organized as follows. Section <ref> includes the notation we follow, and all the relevant dimension notions and their properties. In Section <ref> we prove precise estimates for the box dimension and Assouad spectrum of p-concentric sphere collections, and apply similar dimension-theoretic arguments to prove Theorem <ref>. In Section <ref> we recall the formal definition of quasiconformality, and state the relevant dimension distortion result we need for quasiconformal maps. Moreover, we demonstrate the shared dimension-theoretic properties between polynomially concentric spheres and spiral shells, which allows us to apply the quasiconformal dimension distortion result in order to classify polynomial shells as stated in Theorem <ref>. Section <ref> contains further remarks and potential future directions that relate to this work.



















 §.§ Acknowledgements

The  author wishes to thank Vasiliki Evdoridou, Manisha Garg and Carlos Mudarra for the interesting and motivating discussions on this work.





§ BACKGROUND

We first establish the notation we follow.
Given two positive functions h_1, h_2:(0,∞)→ (0,∞), we write h_1(r) ≲ h_2(r) if there is C>0, independent of r, such that h_1(r)≤ C h_2(r), for all r>0. We call C>0 the comparability constant of the relation h_1(r) ≲ h_2(r).
We write h_1(r) ≳ h_2(r) if h_2(r) ≲ h_1(r). Lastly, we write h_1(r) ≃ h_2(r) if both h_1(r) ≲ h_2(r) and h_1(r) ≳ h_2(r) hold. Note that h_1(r) ≃ h_2(r) in general implies that there are C_1, C_2>0, potentially distinct, such that C_1 h_1(r)≤ h_2(r)≤ C_2 h_1(r) for all r>0. However, this implies that there is a single C=max{ C_1^-1, C_2 }>0 such that C^-1 h_1(r)≤ h_2(r)≤ C h_1(r) for all r>0. In this case, we say that C is the comparability constant of the relation h_1(r) ≃ h_2(r). 
Given x∈^d, r>0, we denote by B(x,r) the open ball centered at x of radius r, and by S(x,r)=∂ B(x,r) the corresponding sphere in ^d. Recall that a set S⊂^d is called a topological sphere, if there is a homeomorphism f:^d→^d with S=f(S(0,1)).
Given two compact sets A,B⊂^d, we denote their usual distance by (A,B), and their Hausdorff distance by

    d_H(A,B)=max{sup{({a},B):  a∈ A}, sup{(A,{b}):  b∈ B}}.




 
We next define the dimension notions we need, along with their properties. We refer to the books by Falconer <cit.> and Fraser <cit.> for thorough expositions on these topics.
Let E ⊂ℝ^d be a non-empty bounded set. For r>0, we denote by N_r (E) the smallest number of r-cubes needed to cover E, i.e., cubes that lie in the axes-oriented r-mesh of ^d and intersect E (see <cit.>).
The lower and upper box dimensions of E are defined by

    _B E = lim inf_ r → 0 log N_r (E)/-log  r    and    _B E = lim sup_ r  → 0 log N_r (E)/-log  r,

respectively.  If the lower and upper box dimensions are equal,  we call the common value the box dimension of E and denote it by _B E. 

Note that we have _B E=d_0 if, and only if, for all ϵ>0 there is r_0=r_0(ϵ)∈ (0,1) such that

    r^-d_0+ϵ≤ N_r(E)≤ r^-d_0-ϵ,
 for all r<r_0. 





Recall that a map g:^d→^d is a similarity if g(x)=c_g 𝐀x+t for all x∈^d, where c_g>0, 𝐀∈𝒪(,d) is a real orthogonal matrix, and t∈^d. We call the scalar c_g the similarity ratio of g. A standard property of the covering number N_r(E) that we need is the way it is distorted by similarities. More specifically, if g is a similarity map with similarity ratio c, then

    N_r(g(E))=N_r/c(E),
 for all r∈ (0,1) (see, for instance, <cit.> Proposition 2.5). Given that _B E=d_0, using the above, along with (<ref>) results in

    (r/c)^-d_0+ϵ≤ N_r(E)≤(r/c)^-d_0-ϵ,
 for all r<min{c r_0, r_0}. This gives a relation between covering numbers of similar topological spheres that is crucial in our arguments in Section <ref>.

For an arbitrary (not necessarily bounded) set E ⊂^d, the Assouad dimension of E is

    _A E = inf{α>0  : ∃ C>0 N_r(B(x,R) ∩ E) ≤ C (R/r)^α}.
 While we do not use the Assouad dimension definition directly, we include it for the sake of completeness, and to relate it to the notion of the Assouad spectrum.

The Assouad spectrum, introduced by Fraser and Yu <cit.>, is a one-parameter family of metrically defined dimensions which interpolates between the upper box dimension and the Assouad dimension. Specifically, the Assouad spectrum of a set E ⊂^d is a collection of values {_A^θ E:0<θ<1}, where 

    _A^θ E=inf{α>0  : ∃ C>0 N_r(B(x,r^θ) ∩ E) ≤ C (r^θ/r)^α},
 for all θ∈ (0,1). We may also refer to the value _A^θ E as the (θ)-Assouad spectrum of E, for a specific  θ∈ (0,1).

The Assouad spectrum essentially captures the growth rate of the covering number N_r(B(x,R)∩ E) for scales 0<r≤ R<1 related by R = r^θ. The map θ↦_A^θ E is continuous at all 0<θ<1  <cit.>, and for a fixed bounded set E⊂^d we have

    _A^θ E → E   ,     _A^θ E →_qAE     ,
 
where _qAE denotes the quasi-Assouad dimension of E, a variant of Assouad dimension introduced by Lü and Xi <cit.>. We always have _qA E ≤_A E, and equality holds in many situations, including all sets of interest in this paper. We refer to <cit.> for proofs of the above properties.

In Section <ref> we use a slightly modified version of the Assouad spectrum, where the relationship R=r^θ between the two scales of the Assouad dimension definition is relaxed to an inequality R≥ r^θ. Namely, given E⊂^d and θ∈ (0,1), we set

    _A,reg^θ E:=inf{α>0  : ∃ C>0 N_r(B(x,R) ∩ E) ≤ C (R/r)^α},

This modification leads to the notion of upper, or regularized Assouad spectrum, denoted by {_A, reg^θE:0<θ<1}. See <cit.> and <cit.> for more information. Tyson and the author focused on this notion in <cit.>, as it aligns more naturally with properties of the Assouad dimension that are fundamental in dimension distortion results (see also <cit.>). For a fixed θ∈ (0,1) the key relationship between the respective two values in the two spectra (see  <cit.>) is that

    _A, reg^θ E = sup_0<θ'<θ_A^θ'E.
 It is immediate from (<ref>) that if _A^θ E is increasing in θ, then _A, reg^θ E=_A^θ E. This is particularly useful, since _A^θ E is often easier to calculate, while the dimension distortion results we use in Section <ref> are proved for the regularized Assouad spectrum.

Moreover, the relation between the different dimensions for a fixed bounded set E follows by (<ref>) and the respective definitions. Namely,

    E ≤_A^θ E≤_A, reg^θ E ≤ E ≤_A E,
 for all θ∈ (0,1). We also state an upper bound on the Assouad spectrum involving the box dimension, which reduces certain calculations in arguments within Section <ref> (see <cit.>).

    Given a bounded non-empty set E⊂^d with d_0:=_B E, we have
    
    _A^θ E≤min{d_0/1-θ,  d },
 for all θ∈ (0,1).




§ DIMENSIONS AND IMPOSSIBLE CONCENTRIC SPHERES
 


In this section, we first establish certain dimension-theoretic properties for polynomially concentric spheres. Namely, we estimate the box dimension and Assouad spectrum of such collections. We then modify the arguments we employed while proving the aforementioned estimates, in order to show the restrictions of polynomially concentric sphere collections with regard to the generating sphere, which yields Theorem <ref>.

Let p>0 and S_0=f(S(0,1))⊂^d be a topological sphere, for some homeomorphism f:^d→^d, with d_0:=_B S_0≤ d. Note that the topological dimension of S(0,1) is equal to d-1, and, since the topological dimension is preserved under homeomorphisms, so is the topological dimension of S_0. Thus, we immediately have d_0≥ d-1 (see, for instance, <cit.> p. 62). Suppose that
S_p=∪_n=0^∞ S_n is a p-concentric sphere collection centered at some x_0∈^d generated by S_0, as in Definition <ref>. 




We can perform certain reductions on S_p. Without loss of generality, we can assume that x_0=0, since all the dimension notions we consider are translation-invariant. In addition, by finite stability of the box dimension and the Assouad spectrum (see <cit.>, p. 18, 49), we can further assume that for the given S_p, we have n_0=1 in Definition <ref>. The above reductions imply that there is c_1∈ (0,1] such that

    d_H(S_n, {0})=c_1 n^-p,
 for all n∈. In addition,  there is c_2∈ (0,1], independent of n, such that

    (S_n, S_n+1)=c_2(n^-p-(n+1)^-p),
 for all n∈. Moreover, for any n∈, we have that

    (S_n,{0})≃ n^-p,
 where the comparability constant is independent of n. Indeed, the upper bound for (<ref>) follows trivially from (<ref>). In order to prove that (S_n,{0})≳ n^-p, let z_n∈ S_n be a point such that |z_n|=(S_n,{0}). Since S_j for j∈{n+1, n+2, …} are all topological spheres with corresponding topological balls containing 0, the line segment with end-points at z_n and 0 intersects each such S_j, say at z_j∈ S_j. This implies that 

    |z_j-z_j+1|≥(S_j, S_j+1)≃ j^-p-(j+1)^-p,
 for all j∈{n, n+1, …}. However, the sum of all of these distances |z_j-z_j+1| has length less than the line segment where all these points lie on, namely

    (S_n,{0})≳∑_j=n^∞(j^-p-(j+1)^-p)=∑_j=1^∞(j^-p-(j+1)^-p)-∑_j=1^n-1(j^-p-(j+1)^-p).
 Using the fact that for any m∈ we have ∑_j=1^m(j^-p-(j+1)^-p)=1-(1+m)^-p, the proof of (<ref>) is complete. Note that (<ref>) also implies that the similarity ratio of g_n is in fact comparable to n^-p, for all n∈. This allows us to use (<ref>) in our arguments.

The above reductions are enough to start with the calculation of the box dimension of S_p in the case d_0=d-1. While we could replace d_0 in the following proof by the explicit value d-1, we prefer to use the former notation to showcase how similar arguments are used for the proof of Theorem <ref> for d_0>d-1.

 
For a p-concentric sphere collection S_p as outlined above, with d_0=_B S_0=d-1, we have

    _B S_p =  max{d/1+p,  d-1 }.




Fix positive d_1, d_2<d close to d_0 such that d_1<d_0<d_2. Due to _B S_0=d_0 and (<ref>), there is r_0=r_0(d_1,d_2)>0 such that

    r^-d_1≤ N_r(S_0)≤ r^-d_2,
 for all r∈ (0,r_0). Moreover, since f is a homeomorphism, the topological dimension of B_0=f(B(0,1)) is equal to d. It follows that _B B_0=d. Hence, for any tiny ϵ>0, there is r_0'=r_0'(ϵ)>0 such that

    r^-d+ϵ≤ N_r(B_0)≤ r^-d-ϵ,
 for all r∈ (0,r_0').

Fix an arbitrary tiny ϵ>0, and an arbitrary r∈ (0,min{r_0,r_0'}), so that (<ref>), (<ref>) hold, and let n_r  be the unique positive integer satisfying

    n_r^-p-(n_r+1)^-p≤r/c_2d^1/2 < (n_r-1)^-p-n_r^-p.
 In arguments that follow, we make a few more assumptions on how small r is selected to be, which only involve uniform constants, such as p, c_1, c_2, d, and the uniform comparability constant for the similarity ratios of g_j from (<ref>). This ensures there is no loss of generality in these further assumptions on the size of r, while it eases the notation and reduces the number of technical inequalities.





By the choice of n_r in (<ref>), we can split S_p into two disjoint sets, i.e.,  S_p= S_p^+∪ S_p^-, where

    S_p^+:=⋃_n>n_r S_n
  

    S_p^-:= ⋃_n ≤ n_r S_n,

and consider N_r(S_p^+) and N_r(S_p^-) separately. The reason is that r-cubes that intersect S_p^+ are essentially disjoint with those that intersect S_p^- by (<ref>)  and (<ref>), hence resulting in N_r(S_p)=N_r(S_p^+)+N_r(S_p^-). Moreover, by (<ref>) and the right-hand side of (<ref>), the collection of r-cubes intersecting some S_n⊂ S_p^- for n≤ n_r and the collection of r-cubes intersecting some S_m for n≠ m≤ n_r are essentially disjoint collections, implying that N_r(S_p^-)=∑_n=1^n_r N_r(S_n). Moreover, by the left-hand side of (<ref>) and by (<ref>), (<ref>), r is large enough for each cube in the r-mesh intersecting B(0,c_1 n_r^-p) to overlap with more than one of the spheres in S^+, resulting in N_r(S^+)≃ N_r(g_n_r(B_0)), where the comparability constant is independent of r. Therefore, we have

    N_r( S_p )  ≃  N_r(g_n_r(B_0)) + ∑_j=1^n_r  N_r( S_j )

Note that n_r increases as r decreases. Hence, for small enough r (in a way that depends only on p, c_1, c_2, d), the relation (<ref>) actually implies 

    n_r≃ r^-1/p+1,
 where the comparability constant depends only on p, c_1, c_2, d. The similarity ratio of g_n_r is comparable to n_r^-p. Since n_r^p r≃ n_r^-1 by (<ref>), without loss of generality, we can assume that r is small enough, in a way solely dependent on p and the comparability constant in the former relation, so that n_r^p r is smaller than min{r_0,r_0'}. This ensures that (<ref>) can be applied  for the fixed r, which along with (<ref>) and (<ref>) results in

    r^-d-ϵ/1+p≃(1/n_r^p r)^d-ϵ≲ N_r(g_n_r(B_0))≲(1/n_r^p r)^d+ϵ≃ r^-d+ϵ/1+p,
 for the fixed tiny ϵ>0.

Since for any j∈{1,…, n_r} the similarity ratio of g_j is comparable to j^-p, and noting that j^p r≤ n_r^p r<min{r_0,r_0'} and S_j=g_j(S_0), we similarly have by (<ref>)  that

    (r/j^-p)^-d_1≲ N_r(g_j(S_0))≲(r/j^-p)^-d_2.




Combining (<ref>), (<ref>) and (<ref>) results in

    r^-d-ϵ/1+p+∑_j=1^n_rj^-p d_1/r^d_1≲ N_r(S_p) ≲ r^-d+ϵ/1+p+∑_j=1^n_rj^-p d_2/r^d_2.

Suppose that pd_1<1. Then by standard partial sum estimates and (<ref>) we have

    ∑_j=1^n_rj^-p d_1/r^d_1≃ n_r^1-pd_1 r^-d_1≃ r^-1-pd_1/p+1-d_1=r^-1-d_1/p+1.
 Similarly, if pd_1>1, we have

    ∑_j=1^n_rj^-p d_1/r^d_1≃1/pd_1-1 r^-d_1≃ r^-d_1.


A similar case study on the product pd_2 shows that for pd_2<1 we have

    ∑_j=1^n_rj^-p d_2/r^d_2≃ n_r^1-pd_2 r^-d_2≃ r^-1-pd_2/p+1-d_2=r^-1-d_2/p+1,
 and for pd_2>1 we have

    ∑_j=1^n_rj^-p d_2/r^d_2≃1/pd_2-1 r^-d_2≃ r^-d_2.


We now determine _B S_p based on the value of the product pd_0. Suppose pd_0<1, then d_2 can be chosen close enough to d_0 to ensure pd_1<pd_2<1. Thus, by (<ref>), (<ref>), (<ref>) we have

    r^-d-ϵ/1+p+ r^-1-d_1/p+1≲ N_r(S_p)≲ r^-d+ϵ/1+p+ r^-1-d_2/p+1.
 Note that d_2> d-1,  so ϵ can be chosen small enough so that d_2+1> d+ϵ. Since r was arbitrary, the above relation implies that

    1+d_1/p+1≤_B S_p ≤1+d_2/p+1.
 Recall that d_1, d_2 are arbitrary and as close to d_0 as necessary, which combined with the above inequality is enough to show that in the case pd_0<1, we have

    _B S_p =1+d_0/p+1=d/p+1.


On the other hand, if pd_0>1, we similarly make sure pd_2>pd_1>1 by choice of d_1, d_2. Note that by d_2>d-1, the value ϵ can be chosen small enough so that pd_2>1 implies that d+ϵ<pd_2+d_2, which gives

    d+ϵ/p+1<d_2.

Using a similar argument to that of the previous case, and the above inequality, along with (<ref>), (<ref>), (<ref>), we can show that

    d_1≤_B S_p ≤ d_2,
 which is enough to conclude that _B S_p=d_0 as needed.

The case p d_0=1 is treated similarly, using (<ref>), (<ref>), (<ref>), resulting in _B S_p=d_0=d-1, due to the fact that

    1+d_0/1+p= 1+d_0/1+1/d_0=d_0.
 All cases have been addressed and the proof is complete.









	It might be tempting to consider r≃ n_r^-p, so that all but finitely many of the spheres S_j⊂ S_p are covered by the 2^d r-cubes with vertices at 0. However, this choice of n_r would hinder any following covering arguments, and especially the crucial equation N_r(S_p^-)=∑_j=1^n_r N_r(S_j) would no longer be true.




The techniques employed in the proof of Theorem <ref> are in fact very similar to those needed to determine the Assouad spectrum of S_p. By adjusting the right quantitative estimates, we can modify accordingly the arguments within the previous proof, so that they apply in the context of two small scales, r and r^θ, as in the definition of the θ-Assouad spectrum.







    Let S_0⊂^d be a topological sphere and S_p=⋃_n∈_0 S_n be a p-concentric sphere collection generated by S_0 as in Theorem <ref>, with the additional assumption that _A S_0=d-1[Requiring _B S_0=_A S_0 might seem restrictive at first. However, various of those lines of research outlined in the Introduction focus on  d_0-Ahlfors regular spheres (see <cit.> for definition and details), which is a special case of the spheres with _B S_0=_A S_0. Hence, it is a natural condition to consider.]. 
    If p (d-1)≤ 1, then  we have
    
    _A^θ S_p = min{d/(1+p)(1-θ),  d },
 for all θ∈ (0,1). On the other hand, if p (d-1)>1, then
    
    _A^θ S_p = min{ d-1+ θ/p(1-θ),  d},
 for all θ∈ (0,1).





Recall that we assume d_0=d-1=_A S_0. Note that for any θ≥ p/(p+1), both (<ref>) and (<ref>) result in _A^θ S_p=d. Hence, by continuity of the Assouad spectrum as a function of θ, it is enough to show (<ref>) and (<ref>) for all θ<p/(p+1).

Fix θ∈ (0,p/(p+1)) and positive d_1, d_2 close to d_0, such that d_1<d_0<d_2. Moreover, fix r∈ (0,1), ϵ>0 as small as necessary for the following arguments, in a way that depends only on  uniform constants of S_p and ^d, similarly to the proof of Theorem <ref>. Denote by n_r∈ the unique positive integer satisfying

    n_r^-p-(n_r+1)^-p≤r/c_2d^1/2 < (n_r-1)^-p-n_r^-p,
 and by m_r∈ the unique positive integer satisfying

    m_r^-p≤r^θ/c_1 < (m_r-1)^-p.
 Since r was fixed to be small enough, the integers n_r, m_r are large enough, so that by (<ref>) we have

    n_r≃ r^-1/p+1,
 and by (<ref>) we have

    m_r≃ r^-θ/p.
 Due to the choice θ<p/(p+1), by (<ref>) and (<ref>) we also have that m_r≤ n_r. Note that the Assouad spectrum is stable under taking the closure of a set (see <cit.>), which ensures that we may estimate the Assouad spectrum of the closure of S_p,  or simply assume that 0∈ S_p without loss of generality. Moreover, despite the fact that the definition of the Assouad spectrum requires to check balls centered at all points of the given set, we focus on B(0,r^θ) instead, for reasons that become clear later.

Arguing similarly to the proof of Theorem <ref> and splitting B(0,r^θ)∩ S_p into two disjoint sets, we have

    N_r(B(0,r^θ)∩ S_p)≃ N_r(g_n_r(B_0))+∑_j=m_r^n_r N_r(S_j),
 which by (<ref>) and the choices of ϵ, d_1, d_2 implies that

    ∑_j=m_r^n_rj^-pd_1/r^d_1≲ N_r(B(0,r^θ)∩ S_p)≲(n_r^-p/r)^d+ϵ+∑_j=m_r^n_rj^-pd_2/r^d_2.
 Using (<ref>) and standard partial summation techniques on the above relation results in

    r^-d_1max{ m_r^1-pd_1, n_r^1-pd_1}≲ N_r(B(0,r^θ)∩ S_p)≲ r^(p/p+1-1)(d+ϵ)+r^-d_2max{ m_r^1-pd_2, n_r^1-pd_2}.
 The rest of the proof is a case study on the product pd_0, similarly to the proof of Theorem <ref>, while also accounting for the different values of θ that could potentially determine which power of r is dominant.

Suppose p d_0<1. In this case, we only need to find a sequence of balls centered at S_p and bound the corresponding covering number from below, which yields a lower bound on _A^θ S_p. This is because the desired upper bound on _A^θ S_p follows trivially by Theorem <ref> and Lemma <ref>. Hence, we can focus on balls centered at 0 and the lower bound of (<ref>). Namely, by (<ref>) and p d_1<1 we have

    r^-d_1n_r^1-pd_1≲ N_r(B(0,r^θ)∩ S_p),
 which by (<ref>) implies

    N_r(B(0,r^θ)∩ S_p)≳ r^-d_1-1-pd_1/p+1= (r^θ-1)^-d_1(p+1)-(1-pd_1)/(θ-1)(p+1)=(r^θ-1)^d_1+1/(1-θ)(p+1).
 Thus, the above shows that

    _A^θ S_p≥d_1+1/(1-θ)(p+1),
 for arbitrary d_1 close to d_0, which is enough to complete the proof of (<ref>).

Suppose that p d_0>1. Note at this point that if x∈ S_p with |x|≤ 2 r^θ, then

    N_r(B(x,r^θ)∩ S_p)≤ N_r(B(0,10r^θ)∩ S_p)≲ N_r(B(0,r^θ)∩ S_p),
 where the comparability constant only depends on the doubling constant of the space ^d (see for instance <cit.>). Hence, it remains to study the covering number N_r(B(0,r^θ)∩ S_p), and N_r(B(x,r^θ)∩ S_p) for arbitrary x∈ S_p with |x|>2 r^θ. In the latter case, we observe that the individual spheres S_j that intersect B(x,r^θ) are not more than m_r in number, due to (<ref>) and (<ref>) (see also <cit.> for a similar argument). Therefore, using the fact that _A S_0=d_0=_B S_0, which implies that _A^θ S_0= d_0 by (<ref>), and a similarity argument similar to (<ref>), we have 

    N_r(B(x,r^θ)∩ S_p)≲ m_r (r^θ-1)^d_2.
 By (<ref>), the above implies that

    N_r(B(x,r^θ)∩ S_p)≲ r^-θ/p(r^θ-1)^d_2= (r^θ-1)^d_2+θ/p(1-θ).

Hence, due to (<ref>) and the above inequality, it is enough to focus on achieving lower and upper bounds for N_r(B(0,r^θ) in the rest of the proof. The relation p d_0>1 implies for (<ref>) that

    r^-d_1m_r^1-pd_1≲ N_r(B(0,r^θ)∩ S_p)≲ r^(p/p+1-1)(d+ϵ)+r^-d_2 m_r^1-pd_2.
 Using (<ref>) in the above relation results in

    r^-d_1r^-θ(1-pd_1)/p≲ N_r(B(0,r^θ)∩ S_p)≲ r^-(d+ϵ)/p+1+r^-d_2 r^-θ(1-pd_2)/p.
 Note that

    -d_1/θ-1-θ(1-pd_1)/p(θ-1)= d_1+θ/p(1-θ),
 and similarly for d_2 instead of d_1, which when applied to the previous inequality yield

    (r^θ-1)^d_1+θ/p(1-θ)≲ N_r(B(0,r^θ)∩ S_p)≲ (r^θ-1)^d+ϵ/(p+1)(1-θ)+(r^θ-1)^d_2+θ/p(1-θ).
 Regarding the dominant term in the upper bound of (<ref>), note that

    d_0+θ/p(1-θ)>d/(p+1)(1-θ)
 if, and only if

    θ< p(pd_0-d_0-d)/(p+1)(pd_0-1)=p/p+1.
 Hence, by choice of θ, and by taking ϵ to be small enough, we have 

    d_2+θ/p(1-θ)>d+ϵ/(p+1)(1-θ),
 which if applied to (<ref>) yields

    (r^θ-1)^d_1+θ/p(1-θ)≲ N_r(B(0,r^θ)∩ S_p)≲ (r^θ-1)^d_2+θ/p(1-θ).
 As a result, by the above and by (<ref>),(<ref>), we have shown that

    d_1+θ/p(1-θ)≤_A^θ S_p≤ d_2+θ/p(1-θ),
 for d_1 and d_2 arbitrarily close to d_0, which is enough to complete the proof in this case as well.

The case p d_0=1 is treated similarly to the above cases. Notice that in this case we have pd_1<1 and 1<pd_2, which yield different dominant terms on the two sides of (<ref>), but the arguments are identical for each side to the corresponding cases; namely, the lower bound is treated using pd_1<1 as in the case pd_0<1, and the upper bound is treated using pd_2>1 as in the case pd_0>1. This is enough to complete the proof



We emphasize that in the case p(d-1)>1, the Assouad spectrum is not given by the upper bound in Lemma <ref> for 0<θ<p/1+p. This provides us with a family of such examples in every Euclidean space ^d.
	
	


Theorem <ref> in fact implies what the box dimension of S_p is equal to as well, by letting θ→ 0 (see Section <ref>). We note that for p>(d-1)^-1, the box dimension of S_p is trivial, in the sense that _B S_p=d-1. On the other hand, the Assouad spectrum provides non-trivial information on S_p, even in the case where the box dimension fails to do so. 
Moreover, (<ref>), (<ref>)  yield that the Assouad dimension of S_p is equal to d, which also fails to provide additional geometric information compared to the Assouad spectrum. This is an immediate implication of (<ref>). 
 
For all p>0, we have  _A S_p =  S_p = d, where S_p is as in Theorem <ref>. 


We finish this section by showing that a topological sphere of large box dimension cannot generate a  polynomially concentric sphere collection.


Assume towards contradiction that S_p is a p-polynomially concentric sphere collection, as described in the beginning of Section <ref>, but with d_0=_B S_0>d-1. Note that we do not require S_0 to satisfy _A S_0 =d_0. We arrive at a contradiction by providing an absurd lower bound on the Assouad spectrum of S_p for certain θ∈ (0,1).

Fix d_1<d_0 close to d_0, such that d_1>d-1, and θ∈ (0,p/p+1). Moreover, similarly to the proof of Theorem <ref>, fix small r∈ (0,1) and integers  n_r, m_r so that (<ref>), (<ref>) are satisfied. Choosing r small enough ensures that n_r≃ r^-1/p+1 and m_r≃ r^-θ/p, which by choice of θ<p/(p+1) also ensures that m_r≤ n_r. Arguing exactly as in the proof of Theorem <ref> yields

    N_r(B(0,r^θ)∩ S_p)≳∑_j=m_r^n_rj^-pd_1/r^d_1≳ r^-d_1max{ m_r^1-pd_1, n_r^1-pd_1}.
 

Suppose pd_0>1. Then d_1 can also be chosen so that p d_1>1, which implies by (<ref>) that

    N_r(B(0,r^θ)∩ S_p)≳ (r^θ-1)^d_1+θ/p(1-θ),
 exactly as in the proof of Theorem <ref>. Since r, θ are arbitrary, this implies that

    _A^θ S_p ≥ d_1+θ/p(1-θ),
 which also yields

    _A^θ S_p ≥ d_0+θ/p(1-θ),

because d_1<d_0 can be arbitrarily close to d_0. However, note that d_0+θ/p(1-θ)<d if, and only if, θ<p(d-d_0)/p(d-d_0)+1. Since p(d-d_0)/p(d-d_0)+1<p/p+1, due to d_0>d-1, if we pick

    p(d-d_0)/p(d-d_0)+1<θ<p/p+1,
 then (<ref>) implies that _A^θ S_p> d, which is a contradiction.

On the other hand, if pd_0≤ 1, we have pd_1<1, which implies by (<ref>) that

    N_r(B(0,r^θ)∩ S_p)≳ (r^θ-1)^d_1+1/(1-θ)(p+1).
 This similarly yields

    _A^θ S_p ≥d_0+1/(1-θ)(p+1),
 which by choosing

    d(p+1)-(d_0+1)/d(p+1)<θ<p/p+1,
 provides the contradicting estimate _A^θ S_p>d for this case as well, completing the proof.



    For p<d^-1(1+d_0-d), if we let d_1<d_0 arbitrarily close to d_0 and follow the argument of the proof of Theorem <ref> just for the lower bound of N_r(S_p), we get that _B S_p≥d_0+1/1+p>d, by choice of p, which is a contradiction. Hence, the box dimension of S_p can also trace the impossibility of such a concentric collection, but only for certain values of p. Therefore, the investigation of the Assouad spectrum is truly necessary.





§ QUASICONFORMAL CLASSIFICATION OF SPIRAL SHELLS
 

As mentioned in the Introduction, quasiconformal maps are a natural generalization of conformal maps in higher dimensions. There are multiple ways to express the intuition behind quasiconformality, i.e., the property that infinitesimal balls are mapped onto infinitesimal ellipsoids whose eccentricity is globally bounded by a constant. We state the analytic definition of quasiconformal maps, which relates directly to the dimension distortion result of interest (see <cit.> for a thorough exposition). A homeomorphism f:Ω→Ω' between domains in ^d, d ≥ 2, is said to be K-quasiconformal, for some K≥ 1, if f lies in the local Sobolev space W^1,d_loc(Ω:^d), and the inequality


    |Df|^d ≤ K  Df


holds almost everywhere Ω (with respect to the d-Lebesgue measure). Note that Df denotes the (a.e. defined) differential matrix, and |𝐀|=max{|𝐀(𝐯)|:|𝐯|=1} denotes the operator norm of a matrix 𝐀. Recall that the local Sobolev space W^1,d_loc(Ω:^d) is the space of maps defined in Ω with first-order weak derivatives in L^p_loc(Ω:^d) (see <cit.> for more details).


There are many dimension distortion results, for different analytic and geometric mapping classes related to quasiconformal maps, and regarding different dimension notions. See  <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.> for a non-exhaustive list. In fact, the following theorem proved by Tyson and the author (see <cit.>) is crucial for the proof of Theorem <ref>.














Let F:Ω→Ω' be a K-quasiconformal map between domains Ω,  Ω'⊂^d. For t>0 , set θ(t)=1/(t+1). If E is a non-empty bounded set  with closure E⊂Ω, we have

    ( 1 - d/s) ( 1/_A,reg^θ(t/K)E - 1/d) 
     ≤1/_A,reg^θ(t)F(E) - 1/d≤( 1 - d/s)^-1( 1/_A,reg^θ(Kt)E - 1/d),
 for a fixed s=s(d,K)>d that only depends on the space ^d and the constant K of the map F.



We have all the tools necessary to fully classify the polynomial spiral shells up to quasiconformal equivalence.

One direction is immediate, as there exists a  map F with F(_p)=_q, which is K-quasiconformal for any K≥ p/q. Namely, the radial stretch map F:^d→^d with F(0)=0 and

    F(x) = |x|^q/p-1x,
 for all x∈^d∖{0}, is (p/q)-quasiconformal (see <cit.>), which implies that F is K-quasiconformal for all K≥ p/q. Moreover, it is elementary to show that F maps _p onto _q, as required. 

For the other implication, assume towards contradiction that F:^d→^d is a K-quasiconformal map with K<p/q and F(_p)=_q. We aim to calculate the regularized Assouad spectrum of the shells _p, _q, and use Theorem <ref> to arrive at a contradiction. Set
S_p:= ∪_j=1^∞ S(0,j^-p) and S_q:= ∪_j=1^∞ S(0,j^-q). The sets S_p, S_q are polynomially concentric sphere collections centered at 0 and generated by the unit sphere S(0,1) in ^3. Hence, by Theorem <ref> we have that

    _A^θ S_p ={[               3/(1+p)(1-θ),   if   0<θ <p/p+1,    p≤1/2;                 2+θ/p(1-θ), if    0<θ <p/p+1,    p> 1/2;                          3,             if   p/p+1≤θ<1,;                             ].
 and similarly for _A^θ S_q.


Note that a bi-Lipschitz map h_1 can be constructed in order to map _p^u:=_p∩{ 
(x_1,x_2,x_3):   x_3≥ 0 } onto S_p^u:= S_p∩{ 
(x_1,x_2,x_3):   x_3≥ 0 } (see <cit.> for similar arguments). Similarly, a different bi-Lipschitz map h_2 can be constructed in order to map _p^ℓ:=_p∩{ 
(x_1,x_2,x_3):   x_3≤ 0 } onto S_p^ℓ:= S_p∩{ 
(x_1,x_2,x_3):   x_3≤ 0 }. Using the finite and bi-Lipschitz stability of the Assouad spectrum (see <cit.>, we have

    _A^θ_p = max{_A^θ_p^u, _A^θ_p^ℓ}= max{_A^θ S_p^u, _A^θ S_p^ℓ}= _A^θ S_p,
 for all θ∈ (0,1). Similarly we can show that _A^θ_q= _A^θ S_q. Hence, by the above and (<ref>) we can fully determine the Assouad spectrum of _p and _q. Moreover, since the expressions in (<ref>) are monotone, by the relation of the Assouad spectrum with the regularized Assouad spectrum in (<ref>), we have


    _A,reg^θ_p ={[               3/(1+p)(1-θ),   if   0<θ <p/p+1,    p≤1/2;                 2+θ/p(1-θ), if    0<θ <p/p+1,    p> 1/2;                          3,             if   p/p+1≤θ<1,;                             ].
 and similarly for _A,reg^θ_q.


We have all the necessary dimension-theoretic information in order to apply  Theorem <ref>. In fact, we mostly use the relation between the phase transitions θ_p=p/(p+1) and θ_q=q/(q+1) of the two shells, i.e., the values where the regularized Assouad spectrum reaches the (quasi-)Assouad dimension. Set t=1/q. Then θ(t) = 1/(1+t) = q/(1+q), and so _A^θ(t)(S_q) = 3 by (<ref>). On the other hand,

    θ(t/K) = K/K+t= q K/qK+1 < p/p+1,
 
due to the fact that K<p/q. Therefore, _A^θ(t/K)(S_p) < 3. This leads to a contradiction due to Theorem <ref> and, in particular, due to the left-hand side of (<ref>). Therefore, such a K-quasiconformal map does not exist.




    A closer analysis of the above proof indicates that Theorem <ref> could be generalized to non-global quasiconformal maps. More specifically, the result is still be true if the map F:^d→^d in the statement of the theorem is replaced by a map F:Ω→Ω' between domains of ^d such that _p∪{0}⊂Ω and _q∪{0}⊂Ω'.
    


We emphasize that the Assouad spectrum estimates are crucial in the proof of Theorem <ref>. Indeed, in the case 1/2≤ q≤ p, all other dimension notions do not provide non-trivial information, since _H _p=_B _p=2<3=_A _p, where _H denotes the Hausdorff dimension (see <cit.>), and similarly for _q. The available dimension distortion results for the Hausdorff, box, and Assouad dimension under quasiconformal maps are not enough to classify such polynomial shells. See <cit.> for a more extensive discussion on this phenomenon in the case of polynomial spirals.






§ FINAL REMARKS


Suppose S_0=f(S(0,1))⊂^d is a topological sphere, where f:^d→^d is a homeomorphism, and x_0 is a point lying in the interior of the topological ball f(B(0,1)). We address the existence of a concentric sphere collection generated by S_0, regardless of the value of _B S_0∈ [d-1,d]. Since f(B(0,1)) is an open set, there is a maximal open ball lying in f(B(0,1)), centered at x_0 and of radius r_1>0. Set g_1 to be a similarity such that x_0∈ g_1(B(0,1))⊂ B(x_0,r_1). Repeating the same argument for g_1(B(0,1)) and iterating infinitely many times, we construct a sequence of similarities {g_n}_n∈ that contracts S_0 into the point x_0, i.e., a concentric sphere collection centered at x_0 and generated by S_0. However,  by Theorem <ref>, we have established that the similarity ratios of g_n cannot be comparable to any sequence of the form n^-p, for any p>0, unless _B S_0=d-1. Thus, in the case _B S_0∈ (d-1,d), the sequence a_n in the constructed example has to converge to 0 faster than n^-p does. Note that in such situations, the convergence is typically too fast for any dimension notion to trace  “fractalness" in the collection. For instance, following the method of the proof of Theorem <ref>, it can be shown that a concentric sphere collection with sequence a_n=e^-n has Assouad spectrum (and, similarly, Assouad dimension) equal to _B S_0. 
A natural question motivated by this discussion is to try and fully characterize the topological spheres S_0 from which a polynomially concentric collection can be generated. Given a topological sphere S_0⊂^d, Theorem <ref> shows that _B S_0=d-1 is a necessary condition, but is it sufficient? Namely, if _B S_0=d-1, can we polynomially contract S_0 into a point using similarities? If the answer to this question is positive in general, are there specific restrictions imposed on the degree p of the polynomial rate n^-p, based on properties of S_0?

It should be emphasised that the case p>(d-1)^-1 in Theorem <ref> is another one of the few examples of sets E⊂^d, where _A^θ E is not constant and is not equal to _B E/(1-θ), for θ<1-_B E/d (see Lemma <ref>). Many of the first such examples mostly appeared in ^2, such as the polynomial spirals studied by Fraser in <cit.>, and the elliptical polynomial spirals studied by Falconer-Fraser-Burrell in <cit.> . In fact, the examples studied in the latter paper stand out for one more reason; for fixed p,q, the Assouad spectrum of elliptical spirals _p,q exhibits two distinct phase transitions, i.e., _A^θ_p,q has three different branches as a function, unlike _A^θ_p which only has two (see <cit.>). This motivates the study of another class in ^d, namely that of concentric ellipsoid collections. In particular, this class is defined similarly to the concentric sphere collections in Definition <ref>, but with the second condition (ii) replaced by (S_n, S_n+1)= c_2 b_n, for some strictly decreasing sequence {b_n}_n∈ that converges to 0 at a different rate than {a_n}_n∈. We expect the dimension study of these objects to be quite interesting, potentially providing us with more examples of sets with Assouad spectrum of two distinct phase transition values.




Recall that a homeomorphism f:^d→^d is (α,β)-bi-Hölder continuous on a non-empty set E, with Hölder exponents 0<α≤ 1≤β<∞, if there is a constant C>0 such that

    C^-1|x-y|^β≤ |f(x)-f(y)|≤ C |x-y|^α,
 for all x,y∈ E.
Fraser in <cit.> suggested the following classification-like direction: given two bounded sets X,Y⊂^d, try to construct a bi-Hölder map f such that f(X)=Y. Moreover, determine what is the sharpest possible choice of Hölder exponents for such a map (which depends on properties of X, Y), and compare them to the bounds on the exponents from dimension distortion theorems. We have indirectly stepped towards this direction for spiral shells through Theorem <ref>, and the connection between quasiconformal and bi-Hölder maps.
Namely,  any K-quasiconformal map F:^d→^d is in fact (1/K,K)-bi-Hölder continuous on every compact set E⊂^d (see for instance <cit.>). Hence, in the proof of Theorem <ref>, we provided a (q/p,p/q)-bi-Hölder map that maps _p∪{0} onto _q∪{0}. However, while this map is sharp for the quasiconformal classification of spiral shells, it is not clear whether it is sharp for the bi-Hölder class of maps between such shells. 

One could use the bi-Hölder dimension distortion results for the Assouad spectrum (see <cit.>), and try to come up with bounds on the Hölder exponents of a map that sends _p onto _q in terms of p and q. For instance, if f:^d→^d is (α,β)-bi-Hölder and E⊂^d is bounded,  then  

    _B E/β≤_B f(E)≤_B E/α.

Therefore, in the case 0<q<p<1, using the the box dimension of _p, _q and the above inequality, we arrive at the trivial bound β≥ (q+1)/(p+1), and the bound α≤ (q+1)/(p+1) for the Hölder exponents. This does not necessarily imply that a q+1/p+1-Hölder map f with f(_p)=_q actually exists. Note that for 0<q<p we have

    q/p<q+1/p+1,
 so the latter term is indeed a “better" Hölder exponent, in the sense that it leads to a “smoother" map f.
However, it appears that apart from certain specialized examples, the Hölder dimension distortion estimates are rarely sharp. This was also emphasized in the Hölder unwinding problem for polynomial spirals by Fraser in <cit.>. As a result, it is not unreasonable to suspect that q/p is the sharpest exponent possible, at least for the classification problem involving (α,α^-1)-bi-Hölder maps between spiral shells _p,_q. In fact, a natural direction to explore is the following; suppose the sharp bi-Hölder exponent bounds in a classification problem are not achieved through knowledge of the dimensions of the sets and the Hölder dimension distortion estimates, are the bounds achieved by the corresponding quasiconformal classification results  sharp?



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