Sampling-based proofs of almost-periodicity results and algorithmic applications

We give new and simple combinatorial proofs of almost-periodicity results for sumsets of sets with small doubling in the spirit of Croot and Sisask [7], whose almost-periodicity lemma has had far-reaching implications in additive combinatorics. We provide an alternative point of view which relies only on Chernoff's bound for sampling, and avoids the need for L^p -norm estimates used in the original proof of Croot and Sisask. We demonstrate the usefulness of our new approach by showing that one can easily deduce from it two significant recent results proved using Croot and Sisask almost-periodicity - the quasipolynomial Bogolyubov-Ruzsa lemma due to Sanders [22] and a result on large subspaces contained in sumsets of dense sets due to Croot, Laba and Sisask [6]. We then turn to algorithmic applications, and show that our approach allows for almost-periodicity proofs to be converted in a natural way to probabilistic algorithms that decide membership in almost-periodic sumsets of dense subsets of double-struck F₂ⁿ. Exploiting this, we give a new algorithmic version of the quasipolynomial Bogolyubov-Ruzsa lemma. Together with the results by the last two authors [27], this implies an algorithmic version of the quadratic Goldreich-Levin theorem in which the number of terms in the quadratic Fourier decomposition of a given function, as well as the running time of the algorithm, are quasipolynomial in the error parameter ε. The algorithmic version of the quasipolynomial Bogolyubov-Ruzsa lemma also implies an improvement in running time and performance of the self-corrector for the Reed-Muller code of order 2 at distance 1/2-ε in [27].