A variant of Tao's method with application to restricted sumsets

TL;DR

This paper extends Tao's harmonic analysis approach to restricted sumsets in prime cyclic groups, providing a new proof for a generalized Erdős–Heilbronn type inequality with applications to additive combinatorics.

Contribution

It introduces a modified version of Tao's method to prove a new lower bound for restricted sumsets involving a subset S in prime cyclic groups.

Findings

Established a lower bound for restricted sumsets in Z/pZ involving a subset S.

Extended Tao's harmonic analysis technique to handle restrictions in sumset problems.

Provided a new proof for a generalized Erdős–Heilbronn conjecture.

Abstract

In this paper, we develop Terence Tao's harmonic analysis method and apply it to restricted sumsets. The well known Cauchy-Davenport theorem asserts that if $A$ and $B$ are nonempty subsets of $Z/pZ$ with $p$ a prime, then $|A+B|\ge min{p,|A|+|B|-1}$, where $A+B={a+b: a\in A, b\in B}$. In 2005, Terence Tao gave a harmonic analysis proof of the Cauchy-Davenport theorem, by applying a new form of the uncertainty principle on Fourier transform. We modify Tao's method so that it can be used to prove the following extension of the Erdos-Heilbronn conjecture: If $A,B,S$ are nonempty subsets of $Z/pZ$ with $p$ a prime, then $|{a+b: a\in A, b\in B, a-b not\in S}|\ge min {p,|A|+|B|-2|S|-1}$.