On uniformity of $q$-multiplicative sequences

Abstract

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

25 pages

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Cited by in corpus (3)

• Automatic sequences and generalised polynomials
• Gowers norms for automatic sequences
• Sharp estimates for Gowers norms on discrete cubes