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:ℝ→[−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≤α<1, let ρα∈(0,1/4] be determined by f(ρα)=α, and let Fα(n) denote the infimum of these proportions over all sets of n distinct positive integer frequencies. We prove that Fα(n)≥ 4ραn and that the limit limn→∞Fα(n)1/n exists for every threshold. When ρα−1 is an integer, we obtain Fα(n) &≤ Cαnlog2((1−ρα)/(1−2ρα))ραn, and hence the limit equals ρα. Otherwise, the limit belongs to the interval [ρα,⌊ρα−1⌋−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.
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BibTeX
@misc{giannitsi2026exponential,
title = {{Exponential decay of threshold correlations for symmetric periodic functions}},
author = {Christina Giannitsi and José Madrid},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38135},
eprint = {2610.38135},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38135},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Exponential decay of threshold correlations for symmetric periodic functions
AU - Christina Giannitsi
AU - José Madrid
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38135
UR - https://arxiv.org/abs/2610.38135
AB - 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:ℝ→[−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≤α<1, let ρ_α∈(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)≥ 4ρ_α^n and that the limit lim_n→∞F_α(n)^1/n exists for every threshold. When ρ_α^−1 is an integer, we obtain F_α(n) &≤ C_αn^log_2((1−ρ_α)/(1−2ρ_α))ρ_α^n, and hence the limit equals ρ_α. Otherwise, the limit belongs to the interval [ρ_α,⌊ρ_α^−1⌋^−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.
ER -