A variant of Tao's method with application to restricted sumsets
Abstract
Description
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 $|{a+b: a\in A, b\in B, a-b not\in S}|\ge min {p,|A|+|B|-2|S|-1}$.