Ruoyi Wang

When shifted primes do not occur in difference sets

Let $[N] = \{1,..., N\}$ and let $A$ be a subset of $[N]$. A result of S\'ark\"ozy in 1978 showed that if the difference set $A-A = \{ a - a’: a, a’ \in A\}$ does not contain any number which is one less than a prime, then $A = o(N)$. The quantitative upper bound on $A$ obtained from S\’ark\”ozy’s proof has be improved subsequently by Lucier, and by Ruzsa and Sanders. I will give a brief introduction of the iteration scheme and the Hardy--Littlewood method used in the known proofs, and our major arc estimate which leads to an improved bound.