Arithmetic sensitivity of cumulant growth in lacunary sums: transcendental versus algebraic ratio limits

TL;DR

This paper investigates how the growth of cumulants in lacunary trigonometric sums is highly sensitive to the arithmetic nature of the sequence of frequencies, revealing nuanced differences between transcendental and algebraic ratio limits.

Contribution

It establishes the precise asymptotic behavior of cumulants based on whether the ratio limit is transcendental or algebraic, highlighting the delicate arithmetic dependence.

Findings

Cumulant growth is linear for transcendental ratio limits.

Algebraic ratio limits can lead to different growth orders, such as quadratic growth.

The behavior of cumulants varies significantly with the arithmetic properties of the sequence.

Abstract

We study the asymptotic behavior of cumulants of lacunary trigonometric sums $S_n(\omega) := \sum_{k=1}^n \cos (2 \pi a_k \omega)$, $\omega\in[0,1]$, and show that cumulant growth is highly sensitive to the arithmetic structure of the sequence $(a_k)_{k \geq 1}$ of positive integers. In particular, if $\lim_{k \to \infty} a_{k+1}/a_k = \eta > 1$ for some transcendental number $\eta$, we prove that for every $m\in \mathbb N$ the $m$-th cumulant of $S_n$ is asymptotically equivalent to the $m$-th cumulant of the ``independent model'' $\widetilde{S}_n := \sum_{k=1}^n \cos (2 \pi a_k U_k)$, where $U_1, U_2, \dots$ are independent random variables having uniform distribution on $[0,1]$. In particular, the order of growth of the cumulants as $n \to \infty$ is linear in this case. We also show that the transcendence condition for $\lim_{k \to \infty} a_{k+1}/a_k$ is in general necessary: when the ratio limit $\eta$ is algebraic, the cumulants of $S_n$ may have a different asymptotic order from those of $\widetilde{S}_n$. For instance, for $a_k = 2^k+1$ (with $\eta = 2$), the sixth cumulant of $S_n$ grows quadratically in $n$. In contrast, for $a_k = 2^k$ (again $\eta = 2$) or when $(a_k)_{k \geq 1}$ is the Fibonacci sequence (with $\eta = (1+\sqrt 5)/2$), the $m$-th cumulant of $S_n$ grows linearly as $n\to\infty$, but with a growth rate that differs from the one of the independent model $\widetilde{S}_n$. Overall, our results show that the asymptotic behavior of the cumulants of lacunary trigonometric sums depends on arithmetic effects in a very delicate way. This is particularly remarkable since many other probabilistic limit theorems, such as the Central Limit Theorem, hold for lacunary trigonometric sums in a universal way without any such sensitivity towards arithmetic effects.