Prescribing the binary digits of squarefree numbers and quadratic residues

TL;DR

This paper investigates the distribution of squarefree numbers, quadratic residues, and primitive roots within additive structures, providing new bounds and demonstrating that fixing a small proportion of binary digits does not affect their expected distribution.

Contribution

It introduces novel bounds on the distribution of primitive roots in Hilbert cubes and shows that fixing less than 40% of binary digits preserves the expected count of squarefree numbers.

Findings

Fixing less than 40% of binary digits retains the asymptotic count of squarefree numbers.

Established a new upper bound on the dimension of Hilbert cubes in primitive roots.

Improved the understanding of quadratic residues in sumsets over finite fields.

Abstract

We study the equidistribution of multiplicatively defined sets, such as the squarefree integers, quadratic non-residues or primitive roots, in sets which are described in an additive way, such as sumsets or Hilbert cubes. In particular, we show that if one fixes any proportion less than $40\%$ of the digits of all numbers of a given binary bit length, then the remaining set still has the asymptotically expected number of squarefree integers. Next, we investigate the distribution of primitive roots modulo a large prime $p$, establishing a new upper bound on the largest dimension of a Hilbert cube in the set of primitive roots, improving on a previous result of the authors. Finally, we study sumsets in finite fields and asymptotically find the expected number of quadratic residues and non-residues in such sumsets, given their cardinalities are big enough. This significantly improves on a recent result by Dartyge, Mauduit and S\'ark\"ozy. Our approach introduces several new ideas, combining a variety of methods, such as bounds of exponential and character sums, geometry of numbers and additive combinatorics.