QUADRATIC RESIDUES AND DIFFERENCE SETS: Table 1.

Abstract

It has been conjectured by Sarkozy that with finitely many exceptions, the set of quadratic residues modulo a prime p cannot be represented as a sumset {a + b : a ∈ A, b ∈ B} with non-singleton A,B ⊆ Fp. The case A = B of this conjecture has been recently established by Shkredov. The analogous problem for differences remains open: is it true that for all sufficiently large primes p, the set of quadratic residues modulo p is not of the form {a′ − a′′ : a′, a′′ ∈ A, a′ 6= a′′} with A ⊆ Fp? We attack here a presumably more tractable variant of this problem, which is to show that there is no A ⊆ Fp such that every quadratic residue has a unique representation as a′ − a′′ with a′, a′′ ∈ A, and no non-residue is represented in this form. We give a number of necessary conditions for the existence of such A, involving for the most part the behavior of primes dividing p− 1. These conditions enable us to rule out all primes p in the range 13 1. Shkredov [Sh14] has recently established the particular case B = A of this conjecture, showing that {a′ + a′′ : a′, a′′ ∈ A} 6 = Rp, except if p = 3 and A = {2}. He has also proved that Rp cannot be represented as a restricted sumset : {a′ + a′′ : a′, a′′ ∈ A, a′ 6= a′′} 6 = Rp for A ⊆ Fp, with several exceptions for p ≤ 13. The argument of [Sh14] does not seem to extend to handle differences (instead of sums) and to show that {a′ − a′′ : a′, a′′ ∈ A, a′ 6= a′′} 6 = Rp, A ⊆ Fp. (1) We notice that for equality to hold in (1), one needs to have 2 (|A| 2 ) ≥ |Rp|, which readily yields |A| > √ p/2. (2) 2010 Mathematics Subject Classification. Primary: 11B13; Secondary: 11A15, 11B34, 11P70, 11T21, 05B10.