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Kneser's theorem for upper Buck density and relative results

Francois Hennecart

TL;DR

This work establishes a Kneser-type inverse theorem for the upper Buck density, showing that a strict drop in the Buck density of a sumset forces a modular, largely periodic structure governed by a finite modulus $q$ and a collection of periodic components. It extends the small-doubling analysis to the two-term case, deriving explicit density and structural formulas, and compares Buck-density phenomena with Jin’s upper Banach density results, including counterexamples illustrating distinct modular behavior. The paper also constructs delicate examples and extremal configurations demonstrating that Buck density can behave very differently from classical densities, including cases where $A$ has positive Buck density but $A+A$ lacks a Buck density or has highly uneven Buck-density across residue classes. Collectively, the results illuminate how Buck density interacts with sumsets, periodicity, and modular reductions, offering new inverse-theorem insights and revealing sharp contrasts with Banach-density-based results.

Abstract

Kneser's theorem in the integers asserts that denoting by $ \underline{\mathrm{d}}$ the lower asymptotic density, if $\underline{\mathrm{d}}(X_1+\cdots+X_k)<\sum_{i=1}^k\underline{\mathrm{d}}(X_i)$ then the sumset $X_1+\cdots+X_k$ is \emph{periodic} for some positive integer $q$. In this article we establish a similar statement for upper Buck density and compare it with the corresponding result due to Jin involving upper Banach density. We also provide the construction of sequences verifying counterintuitive properties with respect to Buck density of a sequence $A$ and its sumset $A+A$.

Kneser's theorem for upper Buck density and relative results

TL;DR

This work establishes a Kneser-type inverse theorem for the upper Buck density, showing that a strict drop in the Buck density of a sumset forces a modular, largely periodic structure governed by a finite modulus and a collection of periodic components. It extends the small-doubling analysis to the two-term case, deriving explicit density and structural formulas, and compares Buck-density phenomena with Jin’s upper Banach density results, including counterexamples illustrating distinct modular behavior. The paper also constructs delicate examples and extremal configurations demonstrating that Buck density can behave very differently from classical densities, including cases where has positive Buck density but lacks a Buck density or has highly uneven Buck-density across residue classes. Collectively, the results illuminate how Buck density interacts with sumsets, periodicity, and modular reductions, offering new inverse-theorem insights and revealing sharp contrasts with Banach-density-based results.

Abstract

Kneser's theorem in the integers asserts that denoting by the lower asymptotic density, if then the sumset is \emph{periodic} for some positive integer . In this article we establish a similar statement for upper Buck density and compare it with the corresponding result due to Jin involving upper Banach density. We also provide the construction of sequences verifying counterintuitive properties with respect to Buck density of a sequence and its sumset .

Paper Structure

This paper contains 12 sections, 20 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. General properties of Buck density and first examples
  3. Kneser's theorem for upper Buck density -- Proof of Theorem \ref{['thmprinc']}
  4. Small doubling sets -- Proof of Theorem \ref{['thmsec']}
  5. Banach density versus Buck density
  6. Further results, constructions and concluding remarks
  7. Sparse periodicity
  8. Kneser's theorem with lower Buck density
  9. A startling construction
  10. Completing Proposition \ref{['pp62']}
  11. $A$ has a positive Buck density does not imply that $A+A$ has a Buck density
  12. On a question of Ruzsa

Key Result

Theorem 1.1

Let $k\ge2$ and $X_1,\dots,X_k\subset \mathbb{N}$ be non empty and such that Set $\eta=\eta(X_1,\dots,X_k)>0$ and $\sigma=\sum_{i=1}^k\overline{\mathrm{b}}(X_i)$. Then there exist a positive integer $q\le \frac{2k-2}{\eta\sigma}$ and, for each $1\le i\le k$, a periodic set $A_i=\bigcup_{j=1}^{r_i}(a_{ij}+q\mathbb{N})\in\mathcal{Z}_*$ where the integers $a_{i1},\dots,a_{ir_i}

Theorems & Definitions (43)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Example 2.4
  • Example 2.5
  • ...and 33 more