On uniformity of $q$-multiplicative sequences

TL;DR

This paper proves that oscillating $q$-multiplicative sequences are Gowers uniform of all orders, implying they do not correlate with polynomial phases and serve as good weights in ergodic theorems, with combinatorial implications.

Contribution

It establishes that oscillating $q$-multiplicative sequences are Gowers uniform of all orders, extending Gelfond type properties across all orders.

Findings

Oscillating $q$-multiplicative sequences are Gowers uniform of all orders.

Such sequences do not correlate with polynomial phase functions.

They serve as good weights for ergodic theorems.

Abstract

We show that any $q$-multiplicative sequence which is \emph{oscillating} of order $1$, i.e.\ does not correlate with linear phase functions $e^{2\pi i n\alpha}$ ($\alpha \in \mathbb{R})$, is Gowers uniform of all orders, and hence in particular does not correlate with polynomial phase functions $e^{2\pi i p(n)}$ ($p \in \mathbb{R}[x]$). Quantitatively, we show that any $q$-multiplicative sequence which is of Gelfond type of order 1 is automatically of Gelfond type of all orders. Consequently, any such $q$-multiplicative sequence is a good weight for ergodic theorems. We also obtain combinatorial corollaries concerning linear patterns in sets which are described in terms of sums of digits.