is actually of smaller order than $n^{-1/2}$. To obtain a more concise bound on the discontinuous term, we shall investigate in detail cases where one or more of the ratios $\rho_{j_1j_2}$ converge to an irrational number as $n$ diverges. However, this treatment requires a definition of the “type” of an irrational, and we give that next.

If $x$ is a real number, let $\langle x\rangle$ denote the distance from $x$ to the nearest integer. (In particular, if $\lfloor x\rfloor$ is the integer part function, $\langle x\rangle=\min\{x-\lfloor x\rfloor
,1-(x-\lfloor x\rfloor)\}$.) We say that the irrational number $\rho$ is of type $\eta$ if $\eta$ equals the supremum of all ${\zeta}$ such that $\liminf_{p{{\to}\infty}} p^{\zeta}\langle p\rho\rangle=0$, where $p{{\to}\infty}$ through integer values. Properties of convergents of irrational numbers (specifically, Dirichlet’s Theorem) can be used to prove that the type of any given irrational number always satisfies $\eta\geq1$. It follows from Roth’s Theorem (Roth [@Rot55]) that all algebraic irrationals (that is, all irrational numbers that are roots of polynomials with rational coefficients) are of minimal type, i.e., $\eta=1$, which is one of the cases we consider below.

More generally, if a number is chosen randomly, for example as the value of a random variable having 