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Popular products and continued fractions

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Abstract

We prove bounds for the popularity of products of sets with weak additive structure, and use these bounds to prove results about continued fractions. Namely, considering Zaremba’s set modulo p, that is the set of all a such that \({a \over p} = [{a_1}, \ldots ,{a_s}]\) has bounded partial quotients, aj ⩽ M, we obtain a sharp upper bound for the cardinality of this set.

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Authors and Affiliations

  1. Steklov Mathematical Institute, Gubkina 8, Moscow, 119991, Russia

    Nikolay Moshchevitin & Ilya Shkredov

  2. School of Mathematics, University of Bristol, University Walk, Bristol, BS8 1 TW, UK

    Brendan Murphy

Authors
  1. Nikolay Moshchevitin
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  2. Brendan Murphy
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  3. Ilya Shkredov
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Correspondence to Brendan Murphy.

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Moshchevitin, N., Murphy, B. & Shkredov, I. Popular products and continued fractions. Isr. J. Math. 238, 807–835 (2020). https://doi.org/10.1007/s11856-020-2039-3

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  • DOI: https://doi.org/10.1007/s11856-020-2039-3

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