# Logarithmic Refinements of a Power Weighted Hardy–Rellich-Type Inequality

Fritz Gesztesy  Department of Mathematics, Baylor University, Sid Richardson
Bldg., 1410 S. 4th Street, Waco, TX 76706, USA<EMAIL_ADDRESS>http://www.baylor.edu/math/index.php?id=935340 ,  Michael M. H. Pang
Department of Mathematics, University of Missouri, Columbia, MO 65211, USA
<EMAIL_ADDRESS>https://www.math.missouri.edu/people/pang and  Jonathan
Stanfill  Department of Mathematics, The Ohio State University
100 Math Tower, 231 West 18th Avenue, Columbus, OH 43210, USA
<EMAIL_ADDRESS>https://u.osu.edu/stanfill-13/ Dedicated, with admiration,
to Fedor Sukochev, mathematician extraordinaire

###### Abstract.

The principal purpose of this note is to prove a logarithmic refinement of the
power weighted Hardy–Rellich inequality on $n$-dimensional balls, valid for
the largest variety of underlying parameters and for all dimensions
$n\in{\mathbb{N}}$, $n\geq 2$.

###### Key words and phrases:

Weighted Hardy–Rellich-type inequality, logarithmic refinements.

###### 2020 Mathematics Subject Classification:

Primary: 35A23, 35J30; Secondary: 47A63, 47F05.

To appear in Ann. Funct. Anal.

##  1\. Introduction

In the recent paper [9] we reconsidered the following sharp inequality, first
derived by Caldiroli and Musina [4, Theorem 3.1],

$\displaystyle\begin{split}\int_{{\mathbb{R}}^{n}}|x|^{\gamma}|(\Delta
f)(x)|^{2}\,d^{n}x\geq
C_{n,\gamma}\int_{{\mathbb{R}}^{n}}|x|^{\gamma-4}|f(x)|^{2}\,d^{n}x,&\\\
\gamma\in{\mathbb{R}},\;f\in
C^{\infty}_{0}({\mathbb{R}}^{n}\backslash\\{0\\}),\;n\in{\mathbb{N}},\,n\geq
2,&\end{split}$ (1.1)

where

$C_{n,\gamma}=\min_{j\in{\mathbb{N}}_{0}}\bigg{\\{}\bigg{(}\frac{(n-2)^{2}}{4}-\frac{(\gamma-2)^{2}}{4}+j(j+n-2)\bigg{)}^{2}\bigg{\\}}.$
(1.2)

In addition, we also derived the sharp inequality (sometimes called the
Hardy–Rellich inequality),

$\displaystyle\begin{split}\int_{{\mathbb{R}}^{n}}|x|^{\gamma}|(\Delta
f)(x)|^{2}\,d^{n}x\geq
A_{n,\gamma}\int_{{\mathbb{R}}^{n}}|x|^{\gamma-2}|(\nabla
f)(x)|^{2}\,d^{n}x,&\\\ \gamma\in{\mathbb{R}},\;f\in
C_{0}^{\infty}({\mathbb{R}}^{n}\backslash\\{0\\}),\;n\in{\mathbb{N}},\,n\geq
2,&\end{split}$ (1.3)

where

$A_{n,\gamma}={\min}_{j\in{\mathbb{N}}_{0}}\\{\alpha_{n,\gamma,\lambda_{j}}\\},$
(1.4)

with

$\displaystyle\begin{split}\alpha_{n,\gamma,\lambda_{0}}&=\alpha_{n,\gamma,0}=4^{-1}(n-\gamma)^{2},\\\
\alpha_{n,\gamma,\lambda_{j}}&=\big{[}4^{-1}(n+\gamma-4)(n-\gamma)+\lambda_{j}\big{]}^{2}\Big{/}\big{[}4^{-1}(n+\gamma-4)^{2}+\lambda_{j}\big{]},\quad
j\in{\mathbb{N}}.\end{split}$ (1.5)

In the unweighted case $\gamma=0$ this simplifies to the known fact,

$A_{n,0}=\begin{cases}n^{2}/4,&n\geq 5,\\\ 3,&n=4,\\\ 25/36,&n=3,\\\
0,&n=2.\end{cases}$ (1.6)

In the special case where $C_{n,\gamma}$ in (1.1), or $A_{n,\gamma}$ in (1.3),
vanishes, the resulting inequality is rendered trivial (e.g., there is no
nontrivial inequality of the type (1.3) in the case $n=2$, $\gamma=0$) and
hence one wonders about the possibility of logarithmically refining these
inequalities to prevent them from becoming insignificant.

In this connection we recall that logarithmic refinements of (1.1) were
already known. Indeed, as discussed in [4], whenever,
$4^{-1}\big{[}(\gamma-2)^{2}-(n-2)^{2}\big{]}$ equals one of the eigenvalues
of $-\Delta_{{\mathbb{S}}^{n-1}}$ (i.e., one of the numbers $j(j+n-2)$,
$j\in{\mathbb{N}}_{0}$), then $C_{n,\gamma}$ vanishes, rendering inequality
(1.1) trivial. In this context we recall the following result from [8, Theorem
1.3]:

$\displaystyle\begin{split}&\int_{B_{n}(0;R)}|x|^{\gamma}|(-\Delta
f)(x)|^{2}\,d^{n}x\geq
C_{n,\gamma}\int_{B_{n}(0;R)}|x|^{\gamma-4}|f(x)|^{2}\,d^{n}x\\\
&\quad+\big{\\{}\big{[}(n-\gamma)^{2}+(n+\gamma-4)^{2}\big{]}\big{/}16\big{\\}}\\\
&\qquad\times\int_{B_{n}(0;R)}|x|^{\gamma-4}\Bigg{(}\sum_{k=1}^{N}\prod_{p=1}^{k}[\operatorname{ln}_{p}(\eta/|x|)]^{-2}\Bigg{)}|f(x)|^{2}\,d^{n}x,\\\
&\,R\in(0,\infty),\;\gamma\in{\mathbb{R}},\;N\in{\mathbb{N}},\;\eta\in[e_{N}R,\infty),\;f\in
C_{0}^{\infty}(B_{n}(0;R)\backslash\\{0\\}),\end{split}$ (1.7)

which yields an appropriate logarithmic refinement even if $C_{n,\gamma}$
vanishes. Here $B_{n}(0;R)$ denotes the open ball in ${\mathbb{R}}^{n}$,
$n\in{\mathbb{N}}$, $n\geq 2$, centered at the origin $0$ of radius
$R\in(0,\infty)$, the iterated logarithms $\operatorname{ln}_{k}(\,\cdot\,)$,
$k\in{\mathbb{N}}$, are given by

$\operatorname{ln}_{1}(\,\cdot\,)=\operatorname{ln}(\,\cdot\,),\quad\operatorname{ln}_{k+1}(\,\cdot\,)=\operatorname{ln}\big{(}\operatorname{ln}_{k}(\,\cdot\,)\big{)},\quad
k\in{\mathbb{N}},$ (1.8)

and the iterated exponentials $e_{j}$, $j\in{\mathbb{N}}_{0}$, are introduced
via

$e_{0}=0,\quad e_{j+1}=e^{e_{j}},\quad j\in{\mathbb{N}}_{0}.$ (1.9)

Given the logarithmic refinement (1.7) of (1.1), it is natural to ask if a
corresponding analogous logarithmic refinement of (1.3) exists that prevents
it from becoming insignificant if $A_{n,\gamma}$ vanishes. Answering this
question in the affirmative is the principal purpose of this note. In
particular, we will prove the following inequality in Theorem 2.3:

$\displaystyle\begin{split}&\int_{B_{n}(0;R)}|x|^{\gamma}|(-\Delta
f)(x)|^{2}\,d^{n}x\geq A_{n,\gamma}\int_{B_{n}(0;R)}|x|^{\gamma-2}|(\nabla
f)(x)|^{2}\,d^{n}x\\\
&\quad+4^{-1}\int_{B_{n}(0;R)}|x|^{\gamma-2}\Bigg{(}\sum_{k=1}^{N}\prod_{p=1}^{k}[\operatorname{ln}_{p}(\eta/|x|)]^{-2}\Bigg{)}|(\nabla
f)(x)|^{2}\,d^{n}x\\\
&\quad+4^{-1}\int_{B_{n}(0;R)}|x|^{\gamma-4}\Bigg{(}\sum_{k=1}^{N}\prod_{p=1}^{k}[\operatorname{ln}_{p}(\eta/|x|)]^{-2}\Bigg{)}|(\nabla_{{\mathbb{S}}^{n-1}}f)(x)|^{2}\,d^{n}x,\\\
&\,R\in(0,\infty),\;\gamma\in{\mathbb{R}},\;N,n\in{\mathbb{N}},\,n\geq
2,\;\eta\in[e_{N}R,\infty),\;f\in
C_{0}^{\infty}(B_{n}(0;R)\backslash\\{0\\}).\end{split}$ (1.10)

Once again, this inequality remains meaningful even if $A_{n,\gamma}$
vanishes.

Given the enormity of the literature on (power weighted) Rellich and
Hardy–Rellich-type inequalities, we will not repeat the extensive list (still
necessarily incomplete) of references cited in [9], and so refer the reader to
the latter. However, more specifically, we mention that Caldiroli and Musina
[4] proved in 2012 that the constant $C_{n,\gamma}$ in (1.1) is optimal. (For
various restricted ranges of $\gamma$ see also Adimurthi, Grossi, and Santra
[1], Ghoussoub and Moradifam [10], [11, Sects. 6.3, 6.5, Ch. 7], and Tertikas
and Zographopoulos [13].) The special unweighted case $\gamma=0$ was settled
for $n\geq 5$ by Herbst [12] in 1977 and subsequently by Yafaev [14] in 1999
for $n\geq 3$, $n\neq 4$ (both authors consider much more general fractional
inequalities).

Under various restrictions on $\gamma$, Tertikas and Zographopoulos [13]
obtained in 2007 optimality of $A_{n,\gamma}$ for $n\geq 5$ and
${\mathbb{R}}^{n}$ replaced by appropriate open bounded domains $\Omega$ with
$0\in\Omega$. This is revisited in Ghoussoub and Moradifam [10], [11, Part 2].
Similarly, Tertikas and Zographopoulos [13] obtained optimality of $A_{n,0}$
for $n\geq 5$; Beckner [3] (see also [2]), and subsequently, Ghoussoub and
Moradifam [10], [11, Sects. 6.3, 6.5, Ch. 7] and Cazacu [5], obtained
optimality of $A_{n,0}$ for $n\geq 3$.

As a notational comment we remark that we abbreviate
${\mathbb{N}}_{0}={\mathbb{N}}\cup\\{0\\}$, and denote by
${{\mathbb{S}}^{n-1}}$ the unit sphere in ${\mathbb{R}}^{n}$,
$n\in{\mathbb{N}}$, $n\geq 2$.

##  2\. A logarithmically Modifed Hardy–Rellich-type Inequality

We begin by recalling the following simplified version of [7, Theorem 3.1
$(iii)$].

######  Lemma 2.1.

Let $R\in(0,\infty)$, $\alpha\in{\mathbb{R}}$, $N\in{\mathbb{N}}$,
$\eta\in[e_{N}R,\infty)$, and $f\in C_{0}^{\infty}((0,R))$. Then

$\displaystyle\begin{split}\int_{0}^{R}r^{\alpha}|f^{\prime}(r)|^{2}\,dr&\geq
4^{-1}(1-\alpha)^{2}\int_{0}^{R}r^{\alpha-2}|f(r)|^{2}\,dr\\\
&\quad\,+4^{-1}\int_{0}^{R}r^{\alpha-2}\Bigg{(}\sum_{k=1}^{N}\prod_{p=1}^{k}[\operatorname{ln}_{p}(\eta/r)]^{-2}\Bigg{)}|f(r)|^{2}\,dr,\end{split}$
(2.1)

where the iterated logarithms $\operatorname{ln}_{k}(\,\cdot\,)$,
$k\in{\mathbb{N}}$, are given by

$\operatorname{ln}_{1}(\,\cdot\,)=\operatorname{ln}(\,\cdot\,),\quad\operatorname{ln}_{k+1}(\,\cdot\,)=\operatorname{ln}\big{(}\operatorname{ln}_{k}(\,\cdot\,)\big{)},\quad
k\in{\mathbb{N}},$ (2.2)

and the iterated exponentials $e_{j}$, $j\in{\mathbb{N}}_{0}$, are introduced
via

$e_{0}=0,\quad e_{j+1}=e^{e_{j}},\quad j\in{\mathbb{N}}_{0}.$ (2.3)

###### Proof.

As the current investigation came about while studying factorizations in [9],
we provide a factorization proof of this lemma in the spirit of [9] (see also
[6] for related higher dimensional unweighted factorizations with log
refinements).

Given $R\in(0,\infty)$, $\alpha\in{\mathbb{R}}$, $N\in{\mathbb{N}}$,
$\eta\in[e_{N}R,\infty)$, one defines the differential expression

$T_{N,\alpha}=r^{\alpha/2}\frac{d}{dr}+\frac{\alpha-1}{2}r^{(\alpha-2)/2}+\frac{1}{2}r^{(\alpha-2)/2}\sum_{k=1}^{N}\prod_{p=1}^{k}[\operatorname{ln}_{p}(\eta/r)]^{-1},\quad
r\in(0,R).$ (2.4)

Then, after applying appropriate integration by parts and combining similar
terms, one confirms that for $f\in C_{0}^{\infty}((0,R))$,

$\displaystyle\begin{split}0\leq\int_{0}^{R}|(T_{N,\alpha}f)(r)|^{2}\,dr&=\int_{0}^{R}r^{\alpha}|f^{\prime}(r)|^{2}\,dr-4^{-1}(1-\alpha)^{2}\int_{0}^{R}r^{\alpha-2}|f(r)|^{2}\,dr\\\
&\quad\,-4^{-1}\int_{0}^{R}r^{\alpha-2}\Bigg{(}\sum_{k=1}^{N}\prod_{p=1}^{k}[\operatorname{ln}_{p}(\eta/r)]^{-2}\Bigg{)}|f(r)|^{2}\,dr,\end{split}$
(2.5)

proving (2.1). ∎

Before deriving our next result, we recall some standard notation and facts.
Let $\mathbb{S}^{n-1}$ denote the $(n-1)$-dimensional unit sphere in
${\mathbb{R}}^{n},\ n\in{\mathbb{N}},\ n\geq 2,$ with
$d^{n-1}\omega:=d^{n-1}\omega(\theta)$ the usual volume measure on
$\mathbb{S}^{n-1}$. We denote by $-\Delta_{\mathbb{S}^{n-1}}$ the nonnegative,
self-adjoint Laplace–Beltrami operator in
$L^{2}(\mathbb{S}^{n-1};d^{n-1}\omega)$. Let

$\lambda_{j}=j(j+n-2),\quad j\in{\mathbb{N}}_{0},$ (2.6)

be the eigenvalues of $-\Delta_{{\mathbb{S}}^{n-1}}$, that is,
$\sigma(-\Delta_{{\mathbb{S}}^{n-1}})=\\{j(j+n-2)\\}_{j\in{\mathbb{N}}_{0}}$,
of multiplicity

$m(\lambda_{j})=(2j+n-2)(j+n-2)^{-1}{j+n-2\choose n-2},\quad
j\in{\mathbb{N}}_{0},$ (2.7)

with corresponding eigenfunctions $\varphi_{j,\ell},\ j\in{\mathbb{N}}_{0},\
\ell\in\\{1,\dots,m(\lambda_{j})\\}$. We may (and will) assume that
$\\{\varphi_{j,\ell}\\}_{j\in{\mathbb{N}}_{0},\
\ell\in\\{1,\dots,m(\lambda_{j})\\}}$ is an orthonormal basis of
$L^{2}({{\mathbb{S}}^{n-1}};d^{n-1}\omega)$, and let

$\displaystyle\begin{split}F_{f,j,\ell}(r)=(\varphi_{j,\ell},f(r,\,\cdot\,))_{L^{2}({{\mathbb{S}}^{n-1}};d^{n-1}\omega)}=\int_{{\mathbb{S}}^{n-1}}\overline{\varphi_{j,\ell}(\theta)}f(r,\theta)\,d^{n-1}\omega(\theta),&\\\
f\in C_{0}^{\infty}({\mathbb{R}}^{n}\backslash\\{0\\}),\quad r>0,\quad
j\in{\mathbb{N}}_{0},\ \ell\in\\{1,\dots,m(\lambda_{j})\\}.&\end{split}$ (2.8)

Finally, let $B_{n}(0;R)$ denote the open ball in ${\mathbb{R}}^{n}$,
$n\in{\mathbb{N}}$, $n\geq 2$, centered at the origin $0$ of radius
$R\in(0,\infty)$.

We are now in the position to prove the following lemma which will be combined
with Lemma 2.1 to prove our main result.

######  Lemma 2.2.

Let $R\in(0,\infty)$, $f\in C_{0}^{\infty}(B_{n}(0;R)\backslash\\{0\\})$, and
$g\in C((0,R))$ satisfy $g(r)>0$ for all $r\in(0,R)$. Then

$\int_{B_{n}(0;R)}g(|x|)|(\nabla
f)(x)|^{2}\,d^{n}x=\sum_{j=0}^{\infty}\sum_{\ell=1}^{m(\lambda_{j})}\int_{0}^{R}g(r)\big{[}|F^{\prime}_{f,j,\ell}(r)|^{2}r^{n-1}+\lambda_{j}|F_{f,j,\ell}(r)|^{2}r^{n-3}\big{]}\,dr.$
(2.9)

###### Proof.

We begin by recalling that

$|(\nabla f)(x)|^{2}=|(\partial f/\partial
r)(r,\theta)|^{2}+r^{-2}|(\nabla_{{\mathbb{S}}^{n-1}}f(r,\,\cdot\,))(\theta)|^{2},$
(2.10)

where $\nabla_{{\mathbb{S}}^{n-1}}$ denotes the gradient operator on
${{\mathbb{S}}^{n-1}}$. Thus applying (2.10) and [8, Lemma 2.1] yields

$\displaystyle\int_{B_{n}(0;R)}g(|x|)|(\nabla
f)(x)|^{2}\,d^{n}x=\int_{0}^{R}g(r)\int_{{\mathbb{S}}^{n-1}}|(\nabla
f)(r,\theta)|^{2}\,d^{n-1}\omega(\theta)\,r^{n-1}dr$
$\displaystyle\quad=\int_{0}^{R}g(r)\int_{{\mathbb{S}}^{n-1}}\big{[}|(\partial
f/\partial r)(r,\theta)|^{2}$ $\displaystyle\quad\hskip
76.82234pt+r^{-2}|(\nabla_{{\mathbb{S}}^{n-1}}f(r,\,\cdot\,))(\theta)|^{2}\big{]}\,d^{n-1}\omega(\theta)\,r^{n-1}dr$
$\displaystyle\quad=\int_{0}^{R}g(r)\Bigg{\\{}\int_{{\mathbb{S}}^{n-1}}\Bigg{|}\sum_{j=0}^{\infty}\sum_{\ell=1}^{m(\lambda_{j})}F^{\prime}_{f,j,\ell}(r)\varphi_{j,\ell}(\theta)\Bigg{|}^{2}\,d^{n-1}\omega(\theta)$
$\displaystyle\quad\hskip
51.21504pt+r^{-2}\int_{{\mathbb{S}}^{n-1}}\overline{(-\Delta_{{\mathbb{S}}^{n-1}}f)(r,\theta)}f(r,\theta)\,d^{n-1}\omega(\theta)\Bigg{\\}}\,r^{n-1}dr$
$\displaystyle\quad=\int_{0}^{R}g(r)\Bigg{\\{}\sum_{j=0}^{\infty}\sum_{\ell=1}^{m(\lambda_{j})}|F^{\prime}_{f,j,\ell}(r)|^{2}$
(2.11)
$\displaystyle\qquad\,+r^{-2}\int_{{\mathbb{S}}^{n-1}}\Bigg{[}\sum_{j=0}^{\infty}\sum_{\ell=1}^{m(\lambda_{j})}\lambda_{j}\overline{F_{f,j,\ell}(r)\varphi_{j,\ell}(\theta)}\Bigg{]}$
$\displaystyle\quad\hskip
59.75095pt\times\Bigg{[}\sum_{j=0}^{\infty}\sum_{\ell=1}^{m(\lambda_{j})}F_{f,j,\ell}(r)\varphi_{j,\ell}(\theta)\Bigg{]}\,d^{n-1}\omega(\theta)\Bigg{\\}}\,r^{n-1}dr$
$\displaystyle\quad=\sum_{j=0}^{\infty}\sum_{\ell=1}^{m(\lambda_{j})}\int_{0}^{R}g(r)\big{[}|F^{\prime}_{f,j,\ell}(r)|^{2}+\lambda_{j}r^{-2}|F_{f,j,\ell}(r)|^{2}\big{]}\,r^{n-1}dr,$

proving (2.9) ∎

Explicitly, (2.11) yields

$\displaystyle\begin{split}&\int_{0}^{R}g(r)\int_{{\mathbb{S}}^{n-1}}|(\partial
f/\partial r)(r,\theta)|^{2}\,d^{n-1}\omega(\theta)\,r^{n-1}dr\\\
&\quad=\sum_{j=0}^{\infty}\sum_{\ell=1}^{m(\lambda_{j})}\int_{0}^{R}g(r)|F^{\prime}_{f,j,\ell}(r)|^{2}\,r^{n-1}dr,\end{split}$
(2.12)
$\displaystyle\begin{split}&\int_{0}^{R}g(r)\int_{{\mathbb{S}}^{n-1}}r^{-2}|(\nabla_{{\mathbb{S}}^{n-1}}f(r,\,\cdot\,))(\theta)|^{2}\,d^{n-1}\omega(\theta)\,r^{n-1}dr\\\
&\quad=\sum_{j=0}^{\infty}\sum_{\ell=1}^{m(\lambda_{j})}\lambda_{j}\int_{0}^{R}g(r)r^{-2}|F_{f,j,\ell}(r)|^{2}\,r^{n-1}dr.\end{split}$
(2.13)

The previous results now allow us to prove the main result in this note in the
form of the following Hardy–Rellich-type inequality with logarithmic
refinements.

######  Theorem 2.3.

Let $R\in(0,\infty)$, $\gamma\in{\mathbb{R}}$, $N,n\in{\mathbb{N}}$, with
$n\geq 2$, $\eta\in[e_{N}R,\infty)$, and $f\in
C_{0}^{\infty}(B_{n}(0;R)\backslash\\{0\\})$. Then

$\displaystyle\int_{B_{n}(0;R)}|x|^{\gamma}|(-\Delta f)(x)|^{2}\,d^{n}x\geq
A_{n,\gamma}\int_{B_{n}(0;R)}|x|^{\gamma-2}|(\nabla f)(x)|^{2}\,d^{n}x$
$\displaystyle\quad+4^{-1}\int_{B_{n}(0;R)}|x|^{\gamma-2}\Bigg{(}\sum_{k=1}^{N}\prod_{p=1}^{k}[\operatorname{ln}_{p}(\eta/|x|)]^{-2}\Bigg{)}|(\nabla
f)(x)|^{2}\,d^{n}x$ (2.14)
$\displaystyle\quad+4^{-1}\int_{B_{n}(0;R)}|x|^{\gamma-4}\Bigg{(}\sum_{k=1}^{N}\prod_{p=1}^{k}[\operatorname{ln}_{p}(\eta/|x|)]^{-2}\Bigg{)}|(\nabla_{{\mathbb{S}}^{n-1}}f)(x)|^{2}\,d^{n}x,$

where

$A_{n,\gamma}={\min}_{j\in{\mathbb{N}}_{0}}\\{\alpha_{n,\gamma,\lambda_{j}}\\},$
(2.15)

with

$\displaystyle\begin{split}\alpha_{n,\gamma,\lambda_{0}}&=\alpha_{n,\gamma,0}=4^{-1}(n-\gamma)^{2},\\\
\alpha_{n,\gamma,\lambda_{j}}&=\big{[}4^{-1}(n+\gamma-4)(n-\gamma)+\lambda_{j}\big{]}^{2}\Big{/}\big{[}4^{-1}(n+\gamma-4)^{2}+\lambda_{j}\big{]},\quad
j\in{\mathbb{N}}.\end{split}$ (2.16)

Excluding the cases $(\alpha)$ $n=2,\ \gamma=2$ and $(\beta)$ $n=3,\
\gamma=1$, the constant $A_{n,\gamma}$ on the right-hand side of inequality
(2.14) is optimal.

###### Proof.

By [9, Eq. (A.25)] and [8, Lemmas 2.3 and B.3 $(i)$] one has

$\displaystyle\int_{B_{n}(0;R)}|x|^{\gamma}|(-\Delta f)(x)|^{2}\,d^{n}x$
$\displaystyle\quad=\sum_{j=0}^{\infty}\sum_{\ell=1}^{m(\lambda_{j})}\int_{0}^{R}r^{\gamma+n-1}\big{|}-r^{1-n}\big{[}d/dr\big{(}r^{n-1}F^{\prime}_{f,j,\ell}(r)\big{)}\big{]}+\lambda_{j}r^{-2}F_{f,j,\ell}(r)\big{|}^{2}\,dr$
$\displaystyle\quad=\sum_{j=0}^{\infty}\sum_{\ell=1}^{m(\lambda_{j})}\Bigg{\\{}\int_{0}^{R}r^{\gamma+n-1}\big{|}F^{\prime\prime}_{f,j,\ell}(r)\big{|}^{2}\,dr$
(2.17) $\displaystyle\hskip
85.35826pt+[2\lambda_{j}+(n-1)(1-\gamma)]\int_{0}^{R}r^{\gamma+n-3}\big{|}F^{\prime}_{f,j,\ell}(r)\big{|}^{2}\,dr$
$\displaystyle\hskip
85.35826pt+\lambda_{j}[\lambda_{j}+(\gamma+n-4)(2-\gamma)]\int_{0}^{R}r^{\gamma+n-5}|F_{f,j,\ell}(r)|^{2}\,dr\Bigg{\\}}.$

Furthermore, note that (2.15) implies

$\lambda_{j}A_{n,\gamma}\leq\big{[}4^{-1}(n+\gamma-4)(n-\gamma)+\lambda_{j}\big{]}^{2}-4^{-1}(n+\gamma-4)^{2}A_{n,\gamma},\quad
j\in{\mathbb{N}}_{0},$ (2.18)

or equivalently,

$\lambda_{j}A_{n,\gamma}\leq
4^{-1}(n+\gamma-4)^{2}\big{[}4^{-1}(n-\gamma)^{2}+2\lambda_{j}-A_{n,\gamma}\big{]}+\lambda_{j}[\lambda_{j}+(n+\gamma-4)(2-\gamma)],\quad
j\in{\mathbb{N}}_{0}.$ (2.19)

Applying Lemma 2.1 and (2.19) to (2) yields

$\displaystyle\int_{B_{n}(0;R)}|x|^{\gamma}|(-\Delta
f)(x)|^{2}\,d^{n}x\geq\sum_{j=0}^{\infty}\sum_{\ell=1}^{m(\lambda_{j})}\Bigg{\\{}4^{-1}(2-n-\gamma)^{2}\int_{0}^{R}r^{\gamma+n-3}\big{|}F^{\prime}_{f,j,\ell}(r)\big{|}^{2}\,dr$
$\displaystyle\qquad+4^{-1}\int_{0}^{R}r^{\gamma+n-3}\big{|}F^{\prime}_{f,j,\ell}(r)\big{|}^{2}\Bigg{(}\sum_{k=1}^{N}\prod_{p=1}^{k}[\operatorname{ln}_{p}(\eta/r)]^{-2}\Bigg{)}\,dr$
$\displaystyle\qquad+[2\lambda_{j}+(n-1)(1-\gamma)]\int_{0}^{R}r^{\gamma+n-3}\big{|}F^{\prime}_{f,j,\ell}(r)\big{|}^{2}\,dr$
$\displaystyle\qquad+\lambda_{j}[\lambda_{j}+(\gamma+n-4)(2-\gamma)]\int_{0}^{R}r^{\gamma+n-5}|F_{f,j,\ell}(r)|^{2}\,dr\Bigg{\\}}$
$\displaystyle\quad=\sum_{j=0}^{\infty}\sum_{\ell=1}^{m(\lambda_{j})}\Bigg{\\{}\big{[}4^{-1}(n-\gamma)^{2}+2\lambda_{j}\big{]}\int_{0}^{R}r^{\gamma+n-3}\big{|}F^{\prime}_{f,j,\ell}(r)\big{|}^{2}\,dr$
$\displaystyle\qquad+4^{-1}\int_{0}^{R}r^{\gamma+n-3}\big{|}F^{\prime}_{f,j,\ell}(r)\big{|}^{2}\Bigg{(}\sum_{k=1}^{N}\prod_{p=1}^{k}[\operatorname{ln}_{p}(\eta/r)]^{-2}\Bigg{)}\,dr$
$\displaystyle\qquad+\lambda_{j}[\lambda_{j}+(\gamma+n-4)(2-\gamma)]\int_{0}^{R}r^{\gamma+n-5}|F_{f,j,\ell}(r)|^{2}\,dr\Bigg{\\}}$
$\displaystyle\quad=\sum_{j=0}^{\infty}\sum_{\ell=1}^{m(\lambda_{j})}\Bigg{\\{}A_{n,\gamma}\int_{0}^{R}r^{\gamma+n-3}\big{|}F^{\prime}_{f,j,\ell}(r)\big{|}^{2}\,dr$
$\displaystyle\qquad+4^{-1}\int_{0}^{R}r^{\gamma+n-3}\big{|}F^{\prime}_{f,j,\ell}(r)\big{|}^{2}\Bigg{(}\sum_{k=1}^{N}\prod_{p=1}^{k}[\operatorname{ln}_{p}(\eta/r)]^{-2}\Bigg{)}\,dr$
$\displaystyle\qquad+\big{[}4^{-1}(n-\gamma)^{2}+2\lambda_{j}-A_{n,\gamma}\big{]}\int_{0}^{R}r^{\gamma+n-3}\big{|}F^{\prime}_{f,j,\ell}(r)\big{|}^{2}\,dr$
$\displaystyle\qquad+\lambda_{j}[\lambda_{j}+(\gamma+n-4)(2-\gamma)]\int_{0}^{R}r^{\gamma+n-5}|F_{f,j,\ell}(r)|^{2}\,dr\Bigg{\\}}$
$\displaystyle\quad\geq\sum_{j=0}^{\infty}\sum_{\ell=1}^{m(\lambda_{j})}\Bigg{\\{}A_{n,\gamma}\int_{0}^{R}r^{\gamma+n-3}\big{|}F^{\prime}_{f,j,\ell}(r)\big{|}^{2}\,dr$
$\displaystyle\qquad+4^{-1}\int_{0}^{R}r^{\gamma+n-3}\big{|}F^{\prime}_{f,j,\ell}(r)\big{|}^{2}\Bigg{(}\sum_{k=1}^{N}\prod_{p=1}^{k}[\operatorname{ln}_{p}(\eta/r)]^{-2}\Bigg{)}\,dr$
$\displaystyle\qquad+\big{\\{}4^{-1}(\gamma+n-4)^{2}\big{[}4^{-1}(n-\gamma)^{2}+2\lambda_{j}-A_{n,\gamma}\big{]}$
$\displaystyle\qquad\quad+\lambda_{j}[\lambda_{j}+(\gamma+n-4)(2-\gamma)]\big{\\}}\int_{0}^{R}r^{\gamma+n-5}|F_{f,j,\ell}(r)|^{2}\,dr$
$\displaystyle\qquad+4^{-1}\big{[}4^{-1}(n-\gamma)^{2}+2\lambda_{j}-A_{n,\gamma}\big{]}$
$\displaystyle\qquad\quad\times\int_{0}^{R}r^{\gamma+n-5}\big{|}F_{f,j,\ell}(r)\big{|}^{2}\Bigg{(}\sum_{k=1}^{N}\prod_{p=1}^{k}[\operatorname{ln}_{p}(\eta/r)]^{-2}\Bigg{)}\,dr\Bigg{\\}}$
$\displaystyle\quad\geq\sum_{j=0}^{\infty}\sum_{\ell=1}^{m(\lambda_{j})}\Bigg{\\{}A_{n,\gamma}\int_{0}^{R}\big{[}r^{\gamma+n-3}\big{|}F^{\prime}_{f,j,\ell}(r)\big{|}^{2}+\lambda_{j}r^{\gamma+n-5}|F_{f,j,\ell}(r)|^{2}\big{]}\,dr$
$\displaystyle\qquad+4^{-1}\int_{0}^{R}r^{\gamma-2}\Bigg{(}\sum_{k=1}^{N}\prod_{p=1}^{k}[\operatorname{ln}_{p}(\eta/r)]^{-2}\Bigg{)}\big{[}\big{|}F^{\prime}_{f,j,\ell}(r)\big{|}^{2}r^{n-1}+\lambda_{j}|F_{f,j,\ell}(r)|^{2}r^{n-3}\big{]}\,dr\Bigg{\\}}$
$\displaystyle\qquad+\sum_{j=0}^{\infty}\sum_{\ell=1}^{m(\lambda_{j})}4^{-1}\lambda_{j}\int_{0}^{R}r^{\gamma+n-5}\big{|}F_{f,j,\ell}(r)\big{|}^{2}\Bigg{(}\sum_{k=1}^{N}\prod_{p=1}^{k}[\operatorname{ln}_{p}(\eta/r)]^{-2}\Bigg{)}\,dr,$
(2.20)

where we used the fact that $4^{-1}(n-\gamma)^{2}\geq A_{n,\gamma}$ (following
from letting $j=0$ in (2.15)) in the last inequality. Finally, applying Lemma
2.2 to the last inequality in (2.20) with

$g(r)=r^{\gamma-2}\,\text{ and
}\,g(r)=r^{\gamma-2}\Bigg{(}\sum_{k=1}^{N}\prod_{p=1}^{k}[\operatorname{ln}_{p}(\eta/r)]^{-2}\Bigg{)},\quad
r\in(0,R),$ (2.21)

one obtains, employing (2.12) and (2.13),

$\displaystyle\int_{B_{n}(0;R)}|x|^{\gamma}|(-\Delta f)(x)|^{2}\,d^{n}x\geq
A_{n,\gamma}\int_{B_{n}(0;R)}|x|^{\gamma-2}|(\nabla f)(x)|^{2}\,d^{n}x$
$\displaystyle\quad+4^{-1}\int_{B_{n}(0;R)}|x|^{\gamma-2}\Bigg{(}\sum_{k=1}^{N}\prod_{p=1}^{k}[\operatorname{ln}_{p}(\eta/|x|)]^{-2}\Bigg{)}|(\nabla
f)(x)|^{2}\,d^{n}x$ (2.22)
$\displaystyle\quad+4^{-1}\int_{B_{n}(0;R)}|x|^{\gamma-4}\Bigg{(}\sum_{k=1}^{N}\prod_{p=1}^{k}[\operatorname{ln}_{p}(\eta/|x|)]^{-2}\Bigg{)}|(\nabla_{{\mathbb{S}}^{n-1}}f)(x)|^{2}\,d^{n}x.$

To prove optimality of $A_{n,\gamma}$ (excluding the cases $(\alpha)$ $n=2$,
$\gamma=2$ and $(\beta)$ $n=3$, $\gamma=1$), one can modify the proof of
optimality found in [9, Theorem A.7], and we now recall the major steps of the
latter. That proof begins by choosing a sequence
$\\{f_{m}\\}_{m\in{\mathbb{N}}}\subset C_{0}^{\infty}((0,\infty))$ such that
$f_{m}$ is real-valued and $f_{m}\not\equiv 0$ for all $m\in{\mathbb{N}}$, and

$\lim_{m\to\infty}\left(\int_{0}^{\infty}r^{\gamma+n-1}|f_{m}^{\prime\prime}(r)|^{2}\,dr\right)\left(\int_{0}^{\infty}r^{\gamma+n-5}|f_{m}(r)|^{2}\,dr\right)^{-1}=\frac{(2-\gamma-n)^{2}(4-\gamma-n)^{2}}{16}.$
(2.23)

Next, depending on the values of $\gamma$ and $n$, one chooses an
eigenfunction, $\varphi$, of $-\Delta_{{\mathbb{S}}^{n-1}}$ and defines
$g_{m}\in C_{0}^{\infty}({\mathbb{R}}^{n}\backslash\\{0\\}),\
m\in{\mathbb{N}},$ by

$g_{m}(x)=g_{m}(r,\theta)=f_{m}(r)\varphi(\theta),\quad
x\in{\mathbb{R}}^{n}\backslash\\{0\\}.$ (2.24)

One can then show that

$\dfrac{\int_{{\mathbb{R}}^{n}}|x|^{\gamma}|(-\Delta
g_{m})(x)|^{2}\,d^{n}x}{\int_{{\mathbb{R}}^{n}}|x|^{\gamma-2}|(\nabla
g_{m})(x)|^{2}\,d^{n}x}\underset{m\to\infty}{\longrightarrow}A_{n,\gamma},$
(2.25)

completing the proof of optimality in [9, Theorem A.7]. To modify this proof
for the current purpose of proving optimality of $A_{n,\gamma}$ in (2.14), one
needs to choose a new sequence in $C_{0}^{\infty}((0,R))$ rather than
$C_{0}^{\infty}((0,\infty))$ to begin with. To this end, we choose
$\\{f_{m}\\}_{m\in{\mathbb{N}}}\subset C_{0}^{\infty}((0,\infty))$ as above
and let $f_{m}\in C_{0}^{\infty}((0,\rho_{m}))$ (e.g.,
$\rho_{m}\geq[\sup(\operatorname{supp}(f_{m}))]+1$) for all
$m\in{\mathbb{N}}$. We then define, for all $m\in{\mathbb{N}}$,
$\widehat{f}_{m}\in C_{0}^{\infty}((0,R))$ by

$\widehat{f}_{m}(y)=f_{m}(\rho_{m}y/R),\quad 0<y<R.$ (2.26)

One then readily verifies that

$\lim_{m\to\infty}\left(\int_{0}^{\infty}r^{\gamma+n-1}|\widehat{f}_{m}^{\,\prime\prime}(r)|^{2}\,dr\right)\left(\int_{0}^{\infty}r^{\gamma+n-5}|\widehat{f}_{m}(r)|^{2}\,dr\right)^{-1}=\frac{(2-\gamma-n)^{2}(4-\gamma-n)^{2}}{16}.$
(2.27)

Thus, replacing $\\{f_{m}\\}_{m\in{\mathbb{N}}}\subset
C_{0}^{\infty}((0,\infty))$ by $\\{\widehat{f}_{m}\\}_{m\in{\mathbb{N}}}\in
C_{0}^{\infty}((0,R))$ in the proof of [9, Theorem A.7] shows optimality of
$A_{n,\gamma}$ in (2.14), once again, excluding the cases $(\alpha)$ $n=2,\
\gamma=2$ and $(\beta)$ $n=3,\ \gamma=1$. ∎

######  Remark 2.4.

$(i)$ The proof of Theorem 2.3 is similar to proofs found in [11, Chs. 6, 7],
but due to our application of [7, Theorem 3.1 $(iii)$] in Lemma 2.1, the range
of parameters has now been greatly extended in Theorem 2.3, in particular, the
two-dimensional case $n=2$ in inequality (2.14) appears to have no precedent.
$(ii)$ In [9, Theorems A.5 and A.7] the authors proved Theorem 2.3 without the
log refinement terms (i.e., without the last two terms on the right side of
(2.14)) and for a larger function space
$C^{\infty}_{0}(\mathbb{R}^{n}\backslash\\{0\\})$. But even with this larger
function space and without the log refinement terms, due to the method of
proof, the authors were unable to show optimality of $A_{n,\gamma}$ in the two
excluded cases in Theorem 2.3, that is, for $(\alpha)$ $n=2$ and $\gamma=2$,
and $(\beta)$ $n=3$ and $\gamma=1$. So, the optimality of $A_{n,\gamma}$ for
those two cases remains open.
$(iii)$ We note that the inequality (2.14) was formulated for the smallest
natural function space $f\in C_{0}^{\infty}(B_{n}(0;R)\backslash\\{0\\})$.
Thus, at least in principle, the optimal constants could have increased in the
process when compared to the function spaces $f\in C_{0}^{\infty}(B_{n}(0;R))$
typically employed in [1], [3], [5], [10], [11, Part 2], [13], etc.
Interestingly enough, Theorem 2.3 demonstrates this possible increase in
optimal constants is not happening with $A_{n,\gamma}$. In this context we
note that [11, Ch. 6] derive optimality of $A_{n,\gamma}$ for $f\in
C_{0}^{\infty}(B_{n}(0;R))$.
$(iv)$ Of course, by restriction, the principal inequalities in this paper
(such as (2.14)–(2.16)) extend to the case where $f\in
C_{0}^{\infty}(B_{n}(0;R)\backslash\\{0\\})$, $n\in{\mathbb{N}}$, $n\geq 2$,
is replaced by $f\in C_{0}^{\infty}(\Omega\backslash\\{0\\})$, where
$\Omega\subseteq B_{n}(0;R)$ is open and bounded with $0\in\Omega$, without
changing the constants in these inequalities. $\diamond$

## References

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# On three problems of Y.–G. Chen

Yuchen Ding  (Yuchen Ding) School of Mathematical Science, Yangzhou
University, Yangzhou 225002, People’s Republic of China<EMAIL_ADDRESS>
###### Abstract.

In this short note, we answer two questions of Chen and Ruzsa negatively and
answer a problem of Ma and Chen affirmatively.

###### Key words and phrases:

square–full numbers; composite numbers; Dirichlet’s theorem in arithmetic
progressions; primes; complete sequences

###### 2010 Mathematics Subject Classification:

Primary 11A41.

*Corresponding author

##  1\. Two problems on $r$–full numbers

The irrationalities of certain type numbers are of great interest to
mathematicians. People have known for a long time that $e$ and $\pi$ are
irrational (see e.g. [17]). Following the formulae of Euler involving
Bernoulli’s numbers (see e.g. [17]), we know that

$\zeta(2s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}}$

are irrational numbers for all positive even integers $s$. It wasn’t until
1978 that Apéry [1] gave a miraculous proof for the irrationality of
$\zeta(3)$. For an alternative proof of the irrationality of $\zeta(3)$, one
can also refer to the note of Beukers [2]. In a remarkable paper, Rivoal [18]
showed that there are infinitely many irrational values of the Riemann zeta
function $\zeta(s)$ at odd integers $s$. Rivoal [19] further proved that at
least one of the nine numbers $\zeta(5),\zeta(7),...,\zeta(21)$ is irrational,
which was then sharpen by Zudilin [20] by displaying that at least one of the
four numbers $\zeta(3),\zeta(5),\zeta(7),\zeta(9)$ is irrational.

In another direction, Erdős [10] proved that if $a_{n}\in\mathbb{N}$ and

$\lim_{n\rightarrow\infty}(a_{n+1}-a_{n})=\infty,$

then

$\sum_{n=1}^{\infty}\frac{a_{n}}{2^{a_{n}}}$

is irrational. It is surely that the gaps between the square–free numbers do
not satisfy the above requirement. Thus, Erdős [10] further conjectured that

$\sum_{n~{}\text{square--free}}\frac{n}{2^{n}}$

is also irrational. This conjecture was later resolved by Chen and Ruzsa [8].
Now, let’s fix some terminologies which shall be used in the context below. If
$p^{r}\nmid n$ for any prime $p$, then the integer $n$ is said to be $r$–free.
An integer $n$ is called $r$–full if $p|n$ implies $p^{r}|n$ for any prime
$p$. Conventionally, $2$–free and $2$–full numbers are called square–free and
square–full numbers, respectively. In fact, Chen and Ruzsa [8, Theorem 4]
provided the following more general result.

######  Proposition 1 (Chen–Ruzsa).

Let $r,\ell$ be two integers with $r,\ell\geq 2$ and $\\{a_{n}\\}$ a sequence
of distinct positive integers such that each $a_{n}$ is either $r$–free or
square–full. Then

$\sum_{n=1}^{\infty}a_{n}\ell^{-a_{n}}$

is irrational.

Thinking about their proof, Chen and Ruzsa naturally raised the following two
problems [8, Section 3. Problems].

######  Problem 1.

Let $r,\ell\geq 2$ be two given integers. Then for any $k\geq 1$, is there a
positive integer $m$ such that each of $\ell^{m}+i~{}(1<i<k)$ is $r$–full?

######  Problem 2.

Let $r,\ell\geq 2$ be two given integers. Then for any $k\geq 1$, is there a
positive integer $m$ such that $\ell^{m}+k$ is $r$–full?

Without doubt, Problem 1 implies Problem 2. We shall show that the answer to
Problem 2 is negative, and hence the answer to Problem 1 is negative either.
That is,

######  Theorem 1.

Let $r,\ell\geq 2$ be two given integers. Then there exists a positive integer
$k$ such that $\ell^{m}+k$ is not $r$–full for any positive integer $m$.

To prove Theorem 1, we need the following well-known result due to Dirichlet
(see for example [9, Page 34, Chapter 4]).

######  Lemma 1 (Dirichlet).

Let $a$ and $b$ be integers such that $(a,b)=1$, then there are infinitely
many primes $p\equiv a\ ({\rm{mod}}\ b)$.

###### Proof of Theorem 1.

The proof of our theorem will be divided into three cases.

Case I. Assuming first $\ell=2$, we consider the number $2^{m}+10$. If $m=1$,
then $2^{1}+10=12$ is not $2$–full, hence not $r$-full. If $m\geq 2$, then
$2|2^{m}+10$ but $2^{2}\nmid 2^{m}+10$, which means that $2^{m}+10$ is not
$2$–full for $m\geq 2$, hence not $r$–full either.

Case II. We next assume that there exists a prime $p$ such that $p^{2}|\ell$,
in which case we choose $k=p$. It is sure that $p|\ell^{m}+p$ but
$p^{2}\nmid\ell^{m}+p$. Hence, $\ell^{m}+p$ is not $2$–full for any positive
integers $m$, hence not $r$–full either.

Case III. Finally, we assume that $\ell$ is a square–free number with at least
one odd prime factor $q$. By Dirichlet’s theorem in arithmetic progressions,
i.e., Lemma 1, there is some integer $s\geq 2$ such that $\ell s-1$ is another
prime $q_{*}$, from which we know that $(q_{*},\ell)=1$. Let
$k=\ell(q_{*}-1)$. Then

$\displaystyle\ell^{m}+k=\ell(\ell^{m-1}+q_{*}-1).$ (1)

If $m=1$, then $\ell+k=\ell q_{*}$. Thus we have $q_{*}|\ell+k$ but
$q_{*}^{2}\nmid\ell+k$ since $(q_{*},\ell)=1$. If $m\geq 2$, then
$q|\ell^{m}+k$ by Eq. (1). Recall that $\ell$ is square–free, it follows that
$q^{2}\nmid\ell^{m}+k$ since otherwise we will have

$q|\ell^{m-1}+q_{*}-1$

again by Eq. (1). Thus, we have

$q|q_{*}-1\Rightarrow q|\ell s-2,$

from which we deduce that $q|2$. This is a contradiction with $q\geq 3$.
Therefore, $\ell+k$ is not $2$–full for any $m$, hence not $r$–full either. ∎

##  2\. A problem on complete sequences

Let $\mathbb{N}$ be the set of nonnegative integers. Suppose that $A$ is a
sequence of nonnegative integers whose terms are not necessarily different.
Let $P(A)$ be the set of nonnegative integers which can be represented as the
sum of finitely distinct terms of $A$. The sequence $A$ is said to be complete
if all sufficiently large integers belong to $P(A)$. Conventionally, we
declare that $0\in P(A)$. Burr [5] asked that which subsets $B$ of
$\mathbb{N}$ are equal to $P(A)$ for a given sequence $A$. This problem
remains widely open but some partial results had already been obtained, see
for example [3, 4, 6, 11, 12, 14, 15]. For $S=\\{s_{1},s_{2},s_{3},...\\}$ and
$\alpha>0$, let

$S_{\alpha}=\left\\{\lfloor\alpha s_{1}\rfloor,\lfloor\alpha
s_{2}\rfloor,...\right\\}.$

Typically, Hegyvári [13] proved the following interesting result.

######  Proposition 2 (Hegyvári).

Let $S=\\{s_{1}<s_{2}<\cdot\cdot\cdot\\}$ be a sequence of positive integers
such that

$\lim_{n\rightarrow\infty}(s_{n+1}-s_{n})=\infty$

and $s_{n+1}<\gamma s_{n}$ for all sufficiently large integers $n$, where
$1<\gamma<2$. Suppose that $S_{\alpha}$ is complete for some $\alpha>0$, then
there exists a positive number $\delta$ such that $S_{\beta}$ are complete for
all $\beta\in[\alpha,\alpha+\delta).$

The unnecessary condition

$\lim_{n\rightarrow\infty}(s_{n+1}-s_{n})=\infty$

was then removed by Chen and Fang [7]. In a subsequent article, Ma and Chen
[16] proved that the prerequisite is always valid in Hegyvári’s Theorem.
Motivated by

######  Proposition 3 (Ma–Chen).

Let $S=\\{s_{1}<s_{2}<\cdot\cdot\cdot\\}$ be a sequence of positive integers
such that $s_{n+1}\leq 2s_{n}$ for all sufficiently large integers $n$.
Suppose that $P(S_{\alpha})\neq\mathbb{N}$ for any $0<\alpha<1$. Then for any
positive integer $w$, there exist a real number $0<\alpha<1$ and a positive
integer $n$ such that

$\left\lfloor\alpha s_{n+i}\right\rfloor=2^{i}\quad(i=0,1,...,w).$

Ma and Chen posed the following problem [16, Problem 3.2].

######  Problem 3.

Does there exist a sequence $S=\\{s_{1},s_{2},...\\}$ of positive integers
with $s_{n}<s_{n+1}\leq 2s_{n}$ and a positive integer $m$ such that for each
$0<\alpha<1$,

$\\{2^{i}:0\leq i\leq m\\}\not\subseteq S_{\alpha}?$

They further remarked that $S=\\{n^{2}:n=1,2,...\\}$ is not the required
sequence for their problem. Since Ma and Chen did not indicate any idea or
detail of their claim, the author would like to offer an alternative proof of
it for convenience of the readers. The aim is to prove that for any integer
$m$, there exists $0<\alpha<1$ such that

$\\{2^{i}:0\leq i\leq m\\}\subseteq\left\\{\left\lfloor\alpha
n^{2}\right\rfloor:n=1,2,...\right\\}.$

The argument goes as follows: Let $\alpha$ be the number satisfying the
requirement. For any $0\leq i\leq m$, it suffices to find some $n_{i}$ with
$2^{i}=\left\lfloor\alpha n_{i}^{2}\right\rfloor$, which is equivalent to

$\sqrt{\alpha^{-1}2^{i}}\leq n_{i}<\sqrt{\alpha^{-1}(2^{i}+1)}.$

The above inequalities would be satisfied if we have

$\sqrt{\alpha^{-1}}\left(\sqrt{2^{i}+1}-\sqrt{2^{i}}\right)>1\quad(0\leq i\leq
m).$

The proof ends up with the choice of $\alpha=4^{-1}(2^{m}+1)^{-1}$.

As we mentioned in the abstract, the answer to Problem 3 is affirmative.

######  Theorem 2.

Let

$S=\left\\{\left\lfloor\left(\frac{3}{2}\right)^{n}\right\rfloor:n=1,2,3,...\right\\}.$

Then

$\left\\{2^{3},2^{4}\right\\}\not\subseteq S_{\alpha}$

for any $0<\alpha<1$.

######  Remark 1.

For sufficiently large $n$ we have

$\left\lfloor\left(\frac{3}{2}\right)^{n}\right\rfloor<\left\lfloor\left(\frac{3}{2}\right)^{n+1}\right\rfloor\leq
2\left\lfloor\left(\frac{3}{2}\right)^{n}\right\rfloor.$

So, Theorem 1 gives a positive answer to Problem 3 with a stronger form.
Actually, our theorem can be slightly strengthen with $3/2$ replaced by a
general number $\gamma\in[3/2,2)$, whereas, in the latter case, the exceptions
$2^{3}$ or $2^{4}$ in the theorem should be substituted by two other suitable
integers $2^{j}$ or $2^{j+1}$ with $j$ depending on $\gamma$.

###### Proof of Theorem 2.

Suppose that $2^{3}\in S_{\alpha}$ for some $0<\alpha<1$, then there is a
positive integer $k$ such that

$2^{3}=\left\lfloor\alpha\left\lfloor\left(\frac{3}{2}\right)^{k}\right\rfloor\right\rfloor,$

which means that

$2^{3}\leq\alpha\left\lfloor\left(\frac{3}{2}\right)^{k}\right\rfloor<2^{3}+1,$

and hence

$2^{3}\alpha^{-1}\leq\left\lfloor\left(\frac{3}{2}\right)^{k}\right\rfloor<\left(2^{3}+1\right)\alpha^{-1}.$

Getting removed of the floor function, it follows that

$2^{3}\alpha^{-1}\leq\left(\frac{3}{2}\right)^{k}<\left(2^{3}+1\right)\alpha^{-1}+1,$

from which we deduce that

$\displaystyle
12\alpha^{-1}\leq\left(\frac{3}{2}\right)^{k+1}<\left(12+\frac{3}{2}\right)\alpha^{-1}+\frac{3}{2}$
(1)

and

$\displaystyle
18\alpha^{-1}\leq\left(\frac{3}{2}\right)^{k+2}<\left(18+\frac{9}{4}\right)\alpha^{-1}+\frac{9}{4}.$
(2)

From the right–hand side inequality of (1), we have

$\alpha\left\lfloor\left(\frac{3}{2}\right)^{k+1}\right\rfloor<12+\frac{3}{2}+\frac{3}{2}\alpha<15.$

From the left–hand side inequality of (2), we have

$\alpha\left\lfloor\left(\frac{3}{2}\right)^{k+2}\right\rfloor>\alpha(18\alpha^{-1}-1)>17.$

Thus,

$\left\lfloor\alpha\left\lfloor\left(\frac{3}{2}\right)^{k+1}\right\rfloor\right\rfloor\leq
14\quad\text{whereas}\quad\left\lfloor\alpha\left\lfloor\left(\frac{3}{2}\right)^{k+2}\right\rfloor\right\rfloor\geq
17.$

It follows that $2^{4}\not\in S_{\alpha}$ provided that $2^{3}\in S_{\alpha}$.
∎

## Acknowledgments

The author would like to thank Professor Yong–Gao Chen for his helpful
comments for an early draft.

The author is supported by National Natural Science Foundation of China (Grant
No. 12201544), Natural Science Foundation of Jiangsu Province in China (Grant
No. BK20210784) and China Postdoctoral Science Foundation (Grant No.
2022M710121), the foundations of the projects ”Jiangsu Provincial
Double–Innovation Doctor Program” (Grant No. JSSCBS20211023) and ”Golden
Phoenix of the Green City–Yang Zhou” to excellent PhD (Grant No.
YZLYJF2020PHD051).

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