Combinatorial problems in finite fields and Sidon sets

TL;DR

This paper introduces an elementary approach using Sidon sets to address various combinatorial problems in finite fields, providing alternative proofs and insights without relying on exponential sums.

Contribution

It presents a novel elementary method employing Sidon sets to study sum-product estimates, equation solubility, and sequence distribution in finite fields.

Findings

Derived sum-product estimates using Sidon sets

Established solubility results for certain equations in finite fields

Analyzed distribution of sequences in small intervals

Abstract

We use Sidon sets to present an elementary method to study some combinatorial problems in finite fields, such as sum product estimates, solubility of some equations and distribution of sequences in small intervals. We obtain classic and more recent results avoiding the use of exponential sums, the usual tool to deal with these problems.