Arithmetic Progressions in Sumsets and L^p-Almost-Periodicity

Abstract

Abstract.We prove results about the L p -almost-periodicity of convolutions. One of these follows from a simple but rather general lemma about approximating a sum of functions in L p , and gives a very short proof of a theorem of Green that if A and B are subsets of {1, . . . , N } of sizes αN and βN then A + B contains an arithmetic progression of length at least exp c(αβ log N ) 1/2 − log log N .Another almost-periodicity result improves this bound for densities decreasing with N : we show that under the above hypotheses the sumset A + B contains an arithmetic progression of length at least exp c α log N log 3 2β −1 1/2− log(β −1 log N ) .

Hilbert cubes in arithmetic sets

“…There are a number of results in the same spirit as Lemma 5. Results of Green [22], Croot, Laba and Sisask [6] and Henriot [31] lead to slightly stronger estimates, when k is large. However, it appears, in our application to Theorem 2 this would not even improve the "2" in the the factor 2 + o k (1) so that we did not pursue this path.…”

Roth’s Theorem for Four Variables and Additive Structures in Sums of Sparse Sets

“…See for example [7,15,21,22] for more background. From the perspective of the present paper it is illuminating to consider what is known in this setting for 2A, 3A and 4A, for which the best bounds known for large densities are all due to Sanders.…”

“…Our main tool for showing properties of convolutions is the following L p -almost-periodicity result, which is a version of the main theorem of [10], but with somewhat less detailed moment estimates in the probabilistic arguments; see for example [7,25] for a proof. THEOREM 2.1.…”

“…In both proofs, we shall use the following bootstrapped version of Theorem 2.1, which is a specialization of [7,Theorem 7.4]. such that, for each t ∈ V ,…”

“…This will be somewhat easier work than in the previous few sections as the arguments are iteration-free and so do not require the machinery associated with regular Bohr sets. However, we do require the analogue of Theorem 3.1 for arbitrary finite abelian groups, this being another specialization of [7 and radius at least α 1/2 /d such that, for each t ∈ T ,…”