On a girth-free variant of the Bourgain-Gamburd machine

We use Bourgain's recent bound for short exponential sums to prove certain independence results related to the distribution of squarefree numbers in arithmetic progressions.

This volume contains the papers presented at the 6th conference on Machines, Computations and Universality (MCU 2013). MCU 2013 was held in Zurich, Switzerland, September 9-11, 2013. The MCU series began in Paris in 1995 and has since been concerned with gaining a deeper understanding of computation through the study of models of general purpose computation. This volume continues in this tradition and includes new simple universal models of computation, and other results that...

We examine statistical properties of the Calkin--Wilf tree and give number-theoretical applications.

In this paper, we will provide a mathematically rigorous computer aided estimation for the exact values and robustness for Gelfond exponent of weighted Thue-Morse sequences. This result improves previous discussions on Gelfond exponent by Gelfond, Devenport, Mauduit, Rivat, S\'{a}rk\"{o}zy and Fan et. al.

We give a broad survey of recent results in Enumerative Combinatorics and their complexity aspects.

In this paper, we present a variant of the Balog-Szemer\'edi-Gowers theorem for the Vinogradov system. We then use our result to deduce a higher degree analogue of the sum-product phenomenon.

About twenty years ago, Green wrote a survey article on the utility of looking at toy versions over finite fields of problems in additive combinatorics. This article was extremely influential, and the rapid development of additive combinatorics necessitated a follow-up survey ten years later, which was written by Wolf. Since the publication of Wolf's article, an immense amount of progress has been made on several central open problems in additive combinatorics in both the fin...

Sidon sets are those sets such that the sums of two of its elements never coincide. They go back to the 30s when Sidon asked for the maximal size of a subset of consecutive integers with that property. This question is now answered in a satisfactory way. Their natural generalization, called B 2 [g] sets and defined by the fact that there are at most g ways (up to reordering the summands) to represent a given integer as a sum of two elements of the set, are much more difficult...

In this paper, we will continue to estmate g_1(n|m) for general n and m.

We consider two kinds of problems: the computation of polynomial and rational solutions of linear recurrences with coefficients that are polynomials with integer coefficients; indefinite and definite summation of sequences that are hypergeometric over the rational numbers. The algorithms for these tasks all involve as an intermediate quantity an integer $N$ (dispersion or root of an indicial polynomial) that is potentially exponential in the bit size of their input. Previous ...