An inverse problem for finite Sidon sets

TL;DR

This paper investigates an inverse problem for Sidon sets, aiming to identify linear forms for which a given set exhibits Sidon properties, thus advancing understanding of their structural characterization.

Contribution

It introduces the inverse problem for Sidon sets, focusing on determining linear forms that make a given set a Sidon set for those forms.

Findings

Characterization of linear forms for Sidon sets

Conditions under which a set is Sidon for a given form

New insights into the structure of Sidon sets

Abstract

Here is a direct problem for Sidon sets: Given a linear form $\varphi = c_1 x_1 + \cdots + c_h x_h $, construct and describe sets $A$ that are Sidon sets for $\varphi$. This paper considers an inverse problem for Sidon sets: Given a set $A$, determine the linear forms $\varphi$ such that $A$ is a Sidon set for $\varphi$.