Sidon sets for linear forms

TL;DR

This paper investigates the existence and properties of Sidon sets related to linear forms in vector spaces, establishing conditions for their existence and analyzing their behavior under perturbations and growth constraints.

Contribution

It characterizes when infinite Sidon sets exist for linear forms based on coefficient subset sums and explores their asymptotic and perturbation properties in various vector spaces.

Findings

Existence of infinite Sidon sets depends on distinct subset sums of coefficients.

Every sequence in a normed space is asymptotic to a Sidon set.

Results on p-adic perturbations and growth bounds of Sidon sets.

Abstract

Let $\varphi(x_1,\ldots, x_h) = c_1 x_1 + \cdots + c_h x_h $ be a linear form with coefficients in a field $\mathbf{F}$, and let $V$ be a vector space over $\mathbf{F}$. A nonempty subset $A$ of $V$ is a $\varphi$-Sidon set if, for all $h$-tuples $(a_1,\ldots, a_h) \in A^h$ and $ (a'_1,\ldots, a'_h) \in A^h$, the relation $\varphi(a_1,\ldots, a_h) = \varphi(a'_1,\ldots, a'_h)$ implies $(a_1,\ldots, a_h) = (a'_1,\ldots, a'_h)$. There exist infinite Sidon sets for the linear form $\varphi$ if and only if the set of coefficients of $\varphi$ has distinct subset sums. In a normed vector space with $\varphi$-Sidon sets, every infinite sequence of vectors is asymptotic to a $\varphi$-Sidon set of vectors. Results on $p$-adic perturbations of $\varphi$-Sidon sets of integers and bounds on the growth of $\varphi$-Sidon sets of integers are also obtained.