Difference sets and the primes

Abstract

Suppose that A is a subset of {1,...,N} such that the difference between any two elements of A is never one less than a prime. We show that |A| = O(N exp(-c(log N)^{1/4})) for some absolute c>0.Comment: 19 pp. Corrected typos. Updated references

Difference sets and the irreducibles in function fields

“…In this section, we shall establish lemmas concerning energy increments. These are analogous to those found in [7,Section 7].…”

“…where log i N denotes i iterations of the log function. To date, the current record upper bound on |A| is due to Ruzsa and Sanders, who showed in [7] that |A| N exp(−c 4 log N ), for some constant c > 0. In view of the many analogies between the integers and the ring F q [t] of polynomials over a finite field F q , it is natural to ask for the analog of Sárközy's theorem in the setting of F q [t].…”

Polynomial configurations in difference sets

“…Similar arguments of this type can be found in Heath-Brown [6] and Szemerédi [14], see also Ruzsa and Sanders [11]. The proof of Proposition 4 will be presented in Sections 3 and 4.…”

Binary Quadratic Forms in Difference Sets

“…Letting E = {n ∈ Z : P + n ⊆ [1, N ]} and F = [−qL, N ] \ E, we see that |F | ≤ 2qL. Therefore, by (18), (19), and the bound 128qL ≤ σN , we have…”