2015/02/24 by Vsevolod F. Lev, Lev, Vsevolod F., Jack Sonn +1 · 3 citations
Engineering · Mathematics · #05B10 #11A15 #11B13 #11B34 #11P70 #11T21 #FOS: Mathematics #Finite Group Theory Research #Limits and Structures in Graph Theory #Number Theory (math.NT) #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.1502.06833
openalex publication_date 2015/02/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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\colon a∈ A, b∈ B\ with non-singleton sets 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"\colon a',a"∈ A, a'≠ 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 uniquerepresentation 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