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Article Dans Une Revue Bulletin of the London Mathematical Society Année : 2019

On uniformity of q-multiplicative sequences

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Résumé

We show that any q-multiplicative sequence which is oscillating of order 1, that is, does not correlate with linear phase functions e2 pi in alpha (alpha is an element of R), is Gowers uniform of all orders, and hence in particular does not correlate with polynomial phase functions e2 pi ip(n) (p is an element of 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.

Dates et versions

hal-03621997 , version 1 (28-03-2022)

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Aihua Fan, Jakub Konieczny. On uniformity of q-multiplicative sequences. Bulletin of the London Mathematical Society, 2019, 51 (3), pp.466-488. ⟨10.1112/blms.12245⟩. ⟨hal-03621997⟩
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