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Triangular gaps in the most frequent sizes of $hA$ for $|A|=4$

Steven Senger

TL;DR

This work investigates triangular gaps in the distribution of the sizes of $hA$ for $A$ chosen as a random 4-element subset of $[1..q]$, with $q$ large relative to $h$. It combines additive combinatorics with a geometric hyperplane-planes approach to bound collision-impacted increases in iterated sumsets, showing that the maximum possible size $M_{h,4}$ occurs most often, while the next few sizes occur with diminishing frequency in a pattern dictated by subtracting tetrahedral numbers ${inom{ ext{ell}+2}{3}}$. A central technical tool is a lemma bounding the size of the set of $A$ that fail to be $B_{h+1}$-sets, together with a geometric argument using planes $P_{oldsymbol{x}}(s)$ in $oldsymbol{R}^4$ to control multiple representations in $(h+1)A$. The results yield precise counts: $|eta^*_{h,4}(q)|=O(h^7 q^3)$ and $|eta^*_{h,4}(q)|= ilde{ heta}(h^{-5} q^3)$, with $(h+1)A$ sizes obeying $|(h+ ext{ell})A| eq M_{h+ ext{ell},4}-{ ext{ell}+2 race 3}$ only rarely, thereby explaining the observed triangular gaps and offering a pathway to generalizations for larger $|A|$.

Abstract

We explain the triangular gaps observed experimentally in the most popular sizes of the $h$-fold iterated sumset, $hA,$ when $A$ is a randomly chosen four-element subset of the first $q$ natural numbers, for $q$ much larger than $h.$

Triangular gaps in the most frequent sizes of $hA$ for $|A|=4$

TL;DR

This work investigates triangular gaps in the distribution of the sizes of for chosen as a random 4-element subset of , with large relative to . It combines additive combinatorics with a geometric hyperplane-planes approach to bound collision-impacted increases in iterated sumsets, showing that the maximum possible size occurs most often, while the next few sizes occur with diminishing frequency in a pattern dictated by subtracting tetrahedral numbers . A central technical tool is a lemma bounding the size of the set of that fail to be -sets, together with a geometric argument using planes in to control multiple representations in . The results yield precise counts: and , with sizes obeying only rarely, thereby explaining the observed triangular gaps and offering a pathway to generalizations for larger .

Abstract

We explain the triangular gaps observed experimentally in the most popular sizes of the -fold iterated sumset, when is a randomly chosen four-element subset of the first natural numbers, for much larger than

Paper Structure

This paper contains 14 sections, 12 theorems, 58 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation
  3. Main results
  4. Acknowledgments
  5. The additive arguments
  6. Proof of Lemma \ref{['gapLemma']}
  7. Proof of Theorem \ref{['4frequent']}:
  8. Proof of Lemma 1 $(ii)$:
  9. Proof of Theorem \ref{['1and3']}:
  10. The geometric arguments (proof of Lemma \ref{['4max']})
  11. Proof of part $(i)$
  12. Proof of Lemma \ref{['4max']} $(ii)$ for small $h$
  13. Proof of part $(iii)$
  14. Generalizing to larger $|A|$

Key Result

Lemma 1

Given $q$ sufficiently large with respect to $h\geq 2,$ we have

Figures (1)

  • Figure 1: Here we use $M_h$ in lieu of $M_{h,4}$ to save space. The numbers indicate which lemmata are used for each estimate for the five largest possible sizes of $hA.$ The size $M_h$ is most frequent, occurring $\Theta(q^4)$ times. The size $M_h-1$ occurs $\Theta(q^3)$ times, by Lemma \ref{['4max']}$(i)$ and $(ii)$, applied to $h-1.$ We also see how Lemma \ref{['4max']} gives upper bounds for the $h$-rare sizes, followed by the contribution due to $B_{h-2}^*$ spiking up at $M_h-4,$ as quantified by Lemma \ref{['4max']}$(i)$ applied to $h-2$ and Lemma \ref{['vecPairLB2']}. We know the gap must be this wide by Lemma \ref{['gapLemma']}.

Theorems & Definitions (20)

  • Lemma 1
  • Theorem 1
  • Lemma 2
  • Theorem 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • ...and 10 more