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arXiv · 1806.04267

On uniformity of $q$-multiplicative sequences

Aihua Fan·Jakub Konieczny
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Abstract

We show that any $q$-multiplicative sequence which is \emph{oscillating} of order $1$, i.e.\ does not correlate with linear phase functions $e^{2πi nα}$ ($α\in \mathbb{R})$, is Gowers uniform of all orders, and hence in particular does not correlate with polynomial phase functions $e^{2πi p(n)}$ ($p \in \mathbb{R}[x]$). 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.

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Aihua Fan, Jakub Konieczny. 2018-12-12. On uniformity of $q$-multiplicative sequences. https://doi.org/10.1112/blms.12245

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