Refined Estimates Concerning Sumsets Contained in the Roots of Unity

Abstract

We prove that the clique number of the Paley graph is at most $\sqrt{p/2}+1$, and that any supposed additive decompositions of the set of quadratic residues can only come from co-Sidon sets.

7 pages. The second version already included a corollary to Theorem 1.2 due to George Shakan that confirms Sarkozy's conjecture for almost all primes

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• On maximal cliques of Cayley graphs over fields
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• Positivity preservers over finite fields
• Multiplicative structure of shifted multiplicative subgroups and its applications to Diophantine tuples
• Exact values and improved bounds on the clique number of cyclotomic graphs
• Restricted sumsets in multiplicative subgroups
• Paley Graphs and Sárközy's Theorem In Function Fields
• Gauss sums and the maximum cliques in generalized Paley graphs of square order
• Quantum computing and the stable set problem
• Multiplicative irreducibility of small perturbations of the set of shifted $k$-th powers
• Induced forests in some distance-regular graphs
• Van Lint-MacWilliams' conjecture and maximum cliques in Cayley graphs over finite fields, II
• Intersective sets over abelian groups
• The subspace structure of maximum cliques in pseudo-Paley graphs from unions of cyclotomic classes