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Higher Convexity and Iterated Sum Sets

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Abstract

Let f be a smooth real function with strictly monotone first k derivatives. We show that for a finite set A, with ∣A + A∣ ≤K∣A∣,

$$\left| {{2^k}f(A) - ({2^k} - 1)f(A)} \right|{ \gg _k}\,{\left| A \right|^{k + 1 - o(1)}}/{K^{{O_k}(1)}}.$$

We deduce several new sum-product type implications, e.g. that A+A being small implies unbounded growth for a many enough times iterated product set A ⋯ A.

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Acknowledgements

Brandon Hanson was supported by the NSF Award 2001622. Oliver Roche-Newton was partially supported by the Austrian Science Fund FWF Projects P 30405-N32 and P 34180. Misha Rudnev is partially supported by the Leverhulme Trust Grant RPG-2017-371. We are grateful to Antal Balog, Peter Bradshaw, Brendan Murphy and Audie Warren for helpful discussions.

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Authors and Affiliations

  1. University of Georgia, Athens, GA, USA

    Brandon Hanson

  2. Johann Radon Institute for Computational and Applied Mathematics, Linz, Austria

    Oliver Roche-Newton

  3. University of Bristol, Bristol, UK

    Misha Rudnev

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  1. Brandon Hanson
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  2. Oliver Roche-Newton
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  3. Misha Rudnev
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Correspondence to Brandon Hanson.

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Hanson, B., Roche-Newton, O. & Rudnev, M. Higher Convexity and Iterated Sum Sets. Combinatorica 42, 71–85 (2022). https://doi.org/10.1007/s00493-021-4578-6

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  • DOI: https://doi.org/10.1007/s00493-021-4578-6

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