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Gowers norms for linearly recurrent numeration systems

Pascal Jelinek

TL;DR

The paper addresses the distribution of digit-sum-like functions in linearly recurrent numeration systems by establishing a Gowers-norm–type bound for systems satisfying Property F. It develops a carry-control framework using a beta-expansion perspective and a Rauzy-fractal-inspired decomposition to enable phase cancellations, and proves an exponential decay bound in depth $\lambda$ for the associated Gowers-type average. This generalizes Gowers-norm results from base representations and Zeckendorf expansions to a broad class of morphic numeration systems, with corollaries for base-$b$ digit sums modulo $m$. The work advances the understanding of higher-order uniformity and correlation properties in non-standard numeration systems and connects to dynamical and morphic sequence theory.

Abstract

Gowers norms have been a key component in the proofs of many breakthrough results in connection to the sum of digits function. Spiegelhofer has used them to show that the Thue-Morse sequence has level of distribution 1 and also that it is equidistributed along cubes. Recently Gowers norms have been used to study the sum of digits function of the Zeckendorf expansion of primes. In this paper we unify the treatments of Gowers norms and give Gowers norms type estimates for a large class of linearly recurrent numeration systems.

Gowers norms for linearly recurrent numeration systems

TL;DR

The paper addresses the distribution of digit-sum-like functions in linearly recurrent numeration systems by establishing a Gowers-norm–type bound for systems satisfying Property F. It develops a carry-control framework using a beta-expansion perspective and a Rauzy-fractal-inspired decomposition to enable phase cancellations, and proves an exponential decay bound in depth for the associated Gowers-type average. This generalizes Gowers-norm results from base representations and Zeckendorf expansions to a broad class of morphic numeration systems, with corollaries for base- digit sums modulo . The work advances the understanding of higher-order uniformity and correlation properties in non-standard numeration systems and connects to dynamical and morphic sequence theory.

Abstract

Gowers norms have been a key component in the proofs of many breakthrough results in connection to the sum of digits function. Spiegelhofer has used them to show that the Thue-Morse sequence has level of distribution 1 and also that it is equidistributed along cubes. Recently Gowers norms have been used to study the sum of digits function of the Zeckendorf expansion of primes. In this paper we unify the treatments of Gowers norms and give Gowers norms type estimates for a large class of linearly recurrent numeration systems.
Paper Structure (4 sections, 5 theorems, 27 equations)

This paper contains 4 sections, 5 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Related works and statements of results
  3. Prerequisites
  4. Proof of the main theorem

Key Result

Theorem 1.4

Let $(G_i)_{i\in \mathbb{N}}$ be a linearly recurrent numeration system satisfying Property F. Then, as $\lambda \to \infty$, we have that for some positive constant $c$, which may depend on $s$. Here we use the usual notation that $e(x)=\exp (2\pi i x)$.

Theorems & Definitions (13)

  • Definition 1.1: Linearly recurrent numeration system
  • Definition 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Corollary 1.6
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 3 more