The following article is a piece of a Master’s research, in which investigates the teaching and learning of arithmetic progression in a learning environment through problem solving. The participants were students from the first year of high school at a private school in Santa Maria / RS. The data collect was developed in eight meetings and used the...
Published in Research in Number Theory
For a given generalized eta-quotient, we show that linear progressions whose residues fulfill certain quadratic equations do not give rise to a linear congruence modulo any prime. This recovers known results for classical eta-quotients, especially the partition function, but also yields linear incongruences for more general weakly holomorphic modul...
Published in Journal d'Analyse Mathématique
Let E ⊆ ℝn be a closed set of Hausdorff dimension α. For m > n, let{B1, …, Bk} be n × (m − n) matrices. We prove that if the system of matrices Bj is non-degenerate in a suitable sense, α is sufficiently close to n, and if E supports a probability measure obeying appropriate dimensionality and Fourier decay conditions, then for a range of m dependi...
Published in The Mathematical Intelligencer
Published in Israel Journal of Mathematics
We investigate the behavior of the divisor function in both short intervals and in arithmetic progressions. The latter problem was recently studied by É. Fouvry, S. Ganguly, E. Kowalski and Ph. Michel. We prove a complementary result to their main theorem. We also show that in short intervals of certain lengths the divisor function has a Gaussian l...
Published in Proceedings of the Steklov Institute of Mathematics
We estimate how many numbers in a given interval have the property that their residues modulo two different fixed numbers belong to two given sets. The estimates obtained are order sharp.
Published in Geometric and Functional Analysis
The celebrated Green-Tao theorem states that there are arbitrarily long arithmetic progressions in the primes. One of the main ingredients in their proof is a relative Szemerédi theorem which says that any subset of a pseudorandom set of integers of positive relative density contains long arithmetic progressions. In this paper, we give a simple pro...
Published in Israel Journal of Mathematics
We establish a version of the Furstenberg-Katznelson multi-dimensional Szemerédi theorem in the primes P:= {2, 3, 5, …}, which roughly speaking asserts that any dense subset of Pd contains finite constellations of any given rational shape. Our arguments are based on a weighted version of the Furstenberg correspondence principle, relative to a weigh...
Published in Lithuanian Mathematical Journal
Published in Journal of Mathematical Sciences
Let P(x) and P3(x) be the error terms in the Gaussian circle problem and the sphere problem, respectively. We investigate the asymptotic behavior of the sums ∑k≤0k≡0modpPk,∑k≤xk≡0modpP3k.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepa...
