Prescribing the binary digits of squarefree numbers and quadratic residues

Rainer Dietmann, Christian Elsholtz, Igor Shparlinski

Research output: Contribution to journalArticlepeer-review


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 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 , 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 that 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.
Original languageEnglish
Pages (from-to)8369-8388
JournalTransactions of the American Mathematical Society
Early online date5 May 2017
Publication statusE-pub ahead of print - 5 May 2017


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