[Paper Review] Gaps in sumsets of $s$ pseudo s-th power sequences

Javier Cilleruelo, Jean‐Marc Deshouillers|arXiv (Cornell University)|Apr 21, 2014

Analytic Number Theory Research3 references3 citations

TL;DR

This paper studies the distribution of gaps in the sumset $ sA $, where $ A $ is a random pseudo $ s $-th power sequence with $ s \geq 2 $. Using probabilistic methods and correlation inequalities, it establishes that almost surely, the lim sup of the normalized gap size $ (b_{n+1} - b_n)/\log b_n $ converges to $ s^s s! / \Gamma^s(1/s) $, providing a precise asymptotic estimate for maximal gap lengths in such sumsets.

ABSTRACT

We study the length of the gaps between consecutive members in the sumset sA when A is a pseudo s-th power sequence, with s>1. We show that, almost surely, limsup (b_{n+1}-b_{n})/log (b_n) = s^s s!/Γ^s(1/s), where b_n are the elements of sA.

Motivation & Objective

• To analyze the distribution of gaps between consecutive elements in the sumset $ sA $, where $ A $ is a pseudo $ s $-th power sequence.
• To determine the almost sure asymptotic behavior of the maximal gap size in $ sA $, normalized by $ \log b_n $.
• To extend probabilistic additive number theory results to higher-order sumsets by rigorously analyzing dependence structures in random representations.
• To establish a sharp upper bound on the lim sup of normalized gap sizes, confirming a heuristic prediction based on Poisson approximation.

Proposed method

• Constructs a probability space where each integer $ n $ is included in $ A $ independently with probability $ \frac{1}{s} n^{-1 + 1/s} $, modeling pseudo $ s $-th powers.
• Defines events $ F_i = \{ sA \cap [i, i + \alpha \log i] = \emptyset \} $ to represent intervals with no elements of $ sA $, modeling potential gaps.
• Applies the generalized Borel-Cantelli lemma to determine whether infinitely many such gap events occur almost surely.
• Uses Janson’s correlation inequality to control dependencies between events $ E_\omega $, where $ \omega $ is a set of integers whose $ s $-fold sums cover a target interval.
• Employs multilinear sum estimates (Lemma 1) to bound probabilities of sum representations and dependencies between overlapping sets.
• Performs a three-way decomposition of the variance-like sum $ \sum_{i<j} (P(F_i \cap F_j) - P(F_i)P(F_j)) $ to verify the Borel-Cantelli condition.

Experimental results

Research questions

• RQ1 What is the almost sure asymptotic behavior of the maximal gap size in the sumset $ sA $ for a pseudo $ s $-th power sequence $ A $?
• RQ2 How does the normalized gap size $ (b_{n+1} - b_n)/\log b_n $ behave as $ n \to \infty $?
• RQ3 Can the heuristic prediction based on Poisson approximation for representation counts be rigorously confirmed for the gap distribution in $ sA $?
• RQ4 What role do dependencies between sum representations play in determining the extremal gap sizes in random sumsets?

Key findings

• The lim sup of the normalized gap size $ (b_{n+1} - b_n)/\log b_n $ converges almost surely to $ \frac{s^s s!}{\Gamma^s(1/s)} $.
• This result confirms a heuristic prediction based on the Poisson distribution of representation counts in $ sA $, with parameter $ \lambda_s = \Gamma^s(1/s)/(s^s s!) $.
• The proof establishes that the probability of infinitely many large gaps of size $ \sim \alpha \log n $ is positive if $ \alpha < \lambda_s^{-1} $, and zero otherwise.
• The analysis rigorously accounts for dependencies between overlapping sum representations using Janson’s inequality and multilinear sum estimates.
• The bound is sharp: no larger normalization would yield a finite lim sup, and the constant matches the inverse of the Poisson parameter $ \lambda_s $.