Exponential decay of threshold correlations for symmetric periodic functions

Abstract

We study how often $n$ distinct integer dilates of a periodic function can all lie above a fixed threshold or all lie below its negative. We consider continuous functions $f:\mathbb R\to[-1,1]$ that are $1$-periodic, even, strictly decreasing on $[0,1/2]$, and satisfy $f(0)=1$ and $f(t+1/2)=-f(t)$. This class includes the cosine and the triangle wave. For $0\leqα<1$, let $ρ_α\in(0,1/4]$ be determined by $f(ρ_α)=α$, and let $F_α(n)$ denote the infimum of these proportions over all sets of $n$ distinct positive integer frequencies. We prove that $F_α(n)\geq 4ρ_α^n$ and that the limit $\lim_{n\to\infty}F_α(n)^{1/n}$ exists for every threshold. When $ρ_α^{-1}$ is an integer, we obtain \begin{align} F_α(n) &\leq C_αn^{\log_2((1-ρ_α)/(1-2ρ_α))}ρ_α^n, \end{align} and hence the limit equals $ρ_α$. Otherwise, the limit belongs to the interval $[ρ_α,\lfloorρ_α^{-1}\rfloor^{-1}]$. The upper bound in the reciprocal case uses a cyclic covering construction previously studied in the context of circular graphs. We give a self-contained proof of the corresponding optimal modulus.