Difference sets and shifted primes

On van der Corput property of shifted primes

“…The bound α(n) = O((log n) −1+o(1) ) = O(exp((−1 + o(1)) log log n)) for the set of shifted primes follows then as a corollary of Theorem 1. This is worse than the bound α(n) = O(exp(−c 4 √ log n)) obtained by Ruzsa and Sanders in [9], but better than earlier bounds in [3] and [10].…”

Multiple recurrence and convergence along the primes

“…(1. 1) Subsequent improvements, first by Lucier [15], and most recently by Ruzsa and Sanders [17], show that the function on the right hand side in the conclusion (1.1) may be replaced by exp(−c(log x) 1/4 ), for some positive absolute constant c. Problems in which one asks for specified constellations of differences between successive terms from a sequence of elements in E, each difference depending on the same shifted prime, have been addressed only very recently. Thus, for example, the problem of exhibiting non-trivial three term arithmetic progressions from E, with common difference a shifted prime, was successfully analysed by Frantzikinakis, Host and Kra [5], with the analogous problem for longer arithmetic progressions conditional on the Inverse Conjecture for Gowers Norms formulated by Green and Tao [8].…”

Quantitative bounds for Gowers uniformity of the Möbius and von Mangoldt functions

“…It was shown by Sárközy [33] that (for N large) any subset of [N ] of size N contains a pattern of the form x, x+p−1 with p a prime. After several improvements [27], [32], the current best known bound quantitative version of this, proved by Wang [40], is that any subset of [N ] of size ≥ N exp(−c(log N ) 1/3 ) contains a pattern of this form. Sárközy's theorem was later generalized to longer progressions by Frantzikinakis-Host-Kra [9], and Wooley-Ziegler [41], who showed that, for any k ≥ 3 and N large enough in terms of k, any subset of [N ] of size N contains a pattern of the form x, x + p − 1, x + 2(p − 1), .…”