Prescribing the binary digits of squarefree numbers and quadratic residues

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 Sárközy. 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.

Upperbound for Dimension of Hilbert Cubes contained in the Quadratic Residues of $\mathbb{F_p}$

“…p 3/19+o(1) from [8], building on earlier works [10,6]. As noted in [6], Theorem 1.2 implies as a special case the Burgess bound g(p) ≤ p 1/4+o (1) on the least primitive root g(p) modulo p [2].…”

On primitive elements of finite fields avoiding affine hyperplanes

“…Since the set S * C contains (q − 1) n elements, such a set contains a primitive element whenever the following inequality holds (q − 1) n + ϕ(q n − 1) > q n . Using a SageMATH program, we directly verify that, with the exception of the pairs (q, n) = (13, 4); (11,4); (11,6); (11,12), the elements of X satisfy the inequality in one of the items (a) or (b) above. This completes the proof of Theorem 1.1.…”

“…as p → +∞. This result is further improved in [4], where milder conditions are imposed on the quantities #D i . The main idea employed in the proof of those results is to provide a nontrivial bound for the sum…”

Sums of algebraic trace functions twisted by arithmetic functions

“…It is also interesting to consider sums of trace functions over integers with other digit restrictions.for example, for integers with fixed binary digits at s prescribed positions, see [12] for some relevant results.…”