A Variant of Tao ’ s Method with Application to Restricted Sumsets

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 ∅ 6= A,B ⊆ Z/pZ with p a prime, then |A+ B| > min{p, |A|+ |B| − 1}, where A + B = {a + b : a ∈ A, b ∈ 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 ErdősHeilbronn conjecture: If A,B, S are nonempty subsets of Z/pZ with p a prime, then ∣