Absolute Bounds for Ergodic Deviations of Linear Form Sequences Relative to Intervals in $\mathbb{T}^1$

TL;DR

This paper establishes explicit upper bounds on the ergodic discrepancy of linear form sequences in the unit interval for almost all vectors, depending on a chosen growth function and convergence conditions.

Contribution

It provides the first explicit bounds on ergodic deviations of linear form sequences relative to intervals, with bounds depending on a convergence condition of a series.

Findings

Bound depends on the growth function ta; ta^{max ext{d,3}}( ext{log log N})

Results hold for full measure set of vectors ta in b R^d

Discrepancy bound is of order (ta( ext{log} N))^d ta^{max ext{d,3}}( ext{log log N})

Abstract

Given a positive increasing function $\varphi$, we show that for a full measure set of vectors $\alpha\in \mathbb{R}^d$, the maximal ergodic discrepancy of the $d$-linear form sequence $\left\{\sum_{1\le i\le d} k_i\alpha_i \mod 1\right\}_{1\le k_i\le N\atop 1\le i\le d}$ relative to intervals in $[0,1)$ has an absolute upper bound of $C(\alpha,\varphi)\cdot (\log N)^d \varphi^{\max\{d,3\}}(\log \log N)$ if $\sum_{n=1}^{\infty} \frac{1}{\varphi(n)}$ converges.