On sumsets and spectral gaps

Suppose that S ⊆ Fp, where p is a prime number. Let λ1, ..., λp be the Fourier coefficients of S arranged as follows |Ŝ(0)| = |λ1| ≥ |λ2| ≥ · · · ≥ |λp|. Then, as is well known, the smaller |λ2| is, relative to |λ1|, the larger the sumset S +S must be; and, one can work out as a function of ε and the density θ = |S|/p, an upper bound for the ratio |λ2|/|λ1| needed in order to guarantee that S + S covers at least (1 − ε)p residue classes modulo p. Put another way, if S has a large spectral gap, then most elements of Fp have the same number of representations as a sum of two elements of S, thereby making S + S large. What we show in this paper is an extension of this fact, which holds for spectral gaps between other consecutive Fourier coefficients λk, λk+1, so long as k is not too large; in particular, our theorem will work so long as 1 ≤ k < log p log 4 Furthermore, we develop results for repeated sums S + S + · · · + S. It is worth noting that this phenomena does not hold in the larger finite field setting Fpn for fixed p, and where we let n → ∞, because, for example, the indicator function for a large subspace of Fpn can have a large spectral gap, and yet the sumset of that subspace with itself equals the subspace (which therefore means it cannot cover density 1− ε fraction of Fpn). The property of Fp that we exploit, which does not hold for Fpn (at least not in the way that Supported in part by an NSF grant. Research partially supported by MNSW grant 2 P03A 029 30