Existence of a Sidon set for the distinct distance constant

TL;DR

The paper addresses the problem of extremizing reciprocal sums over Sidon-type sets, culminating in the existence of Sidon sets that achieve the distinct distance constant (DDC). It develops a compact, variational framework by embedding Sidon sets into a compact space of analytic generating functions $\mathfrak{S}$, with $S\in\mathcal{S}$ iff $f_S\in\mathfrak{S}$, and proves that extremal functionals $\sum s^{-\alpha}$ (for $\alpha>\tfrac{1}{2}$) attain their supremum, in particular yielding a Sidon set with $\sum 1/s = DDC$ and improving the known upper bound $DDC<2.37366$ to $2.247307$. The authors further establish density and approximation results for finite Sidon sets, integrability properties of the generating functions, and a general framework that extends to other patterns such as $B_2[g]$ sequences and sum-free sets, via a general all-patterns theorem guaranteeing existence of maximizers for any continuous functional on a closed pattern. Collectively, these results provide a robust variational method for extremal pattern problems and broaden the scope of maximal-reciprocal-sum phenomena from Sidon sets to a wide class of combinatorial patterns.

Abstract

We highlight a certain compactness of Sidon sets and $B_2[g]$-sets and provide several applications. Notably, we prove the existence of such sets that maximize certain functions. In particular, we show the existence of a Sidon set whose reciprocal sum is equal to the distinct distance constant, which answers a question from Guy and Zhang. We also improve the best known upper bound for this constant.

Theorems & Definitions (33)

• Theorem
• Theorem 1.2
• Theorem 5.1
• Theorem 6.2
• Theorem 6.3
• Corollary 6.4
• Theorem 1.1
• proof
• Theorem 1.2
• proof