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A better than $3/2$ exponent for iterated sums and products over $\mathbb R$

Published online by Cambridge University Press:  10 May 2024

OLIVER ROCHE–NEWTON*
Affiliation:
Institute for Algebra, Johannes Kepler Universität, Altenberger Straβe 69, Linz 4040, Austria. e-mail:o.rochenewton@gmail.com

Abstract

In this paper, we prove that the bound

\begin{equation*}\max \{ |8A-7A|,|5f(A)-4f(A)| \} \gg |A|^{\frac{3}{2} + \frac{1}{54}}\end{equation*}
holds for all $A \subset \mathbb R$, and for all convex functions f which satisfy an additional technical condition. This technical condition is satisfied by the logarithmic function, and this fact can be used to deduce a sum-product estimate
\begin{equation*}\max \{ |16A|, |A^{(16)}| \} \gg |A|^{\frac{3}{2} + c},\end{equation*}
for some $c\gt 0$. Previously, no sum-product estimate over $\mathbb R$ with exponent strictly greater than $3/2$ was known for any number of variables. Moreover, the technical condition on f seems to be satisfied for most interesting cases, and we give some further applications. In particular, we show that
\begin{equation*}|AA| \leq K|A| \implies \,\forall d \in \mathbb R \setminus \{0 \}, \,\, |\{(a,b) \in A \times A : a-b=d \}| \ll K^C |A|^{\frac{2}{3}-c^{\prime}},\end{equation*}
where $c,C \gt 0$ are absolute constants.

Type
Research Article
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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References

Bourgain, J. and Chang, M.-C.. On the size of k-fold sum and product sets of integers. J. Amer. Math. Soc. 17 (2004), 473497.CrossRefGoogle Scholar
Bradshaw, P. J.. Growth in sumsets of higher convex functions. Combinatorica 43(4) (2023), 769–789.CrossRefGoogle Scholar
Elekes, G., Nathanson, M. and Ruzsa, I.. Convexity and sumsets. J Number Theory. 83 (1999), 194201.CrossRefGoogle Scholar
Erdős, P. and Szemerédi, E.. On sums and products of integers. In Studies in Pure Mathematics (Birkhäuser, Basel, 1983), 213–218.CrossRefGoogle Scholar
Hanson, B., Roche–Newton, O. and Rudnev, M.. Higher convexity and iterated sum sets. Combinatorica 42(1) (2022), 71–85.CrossRefGoogle Scholar
Hanson, B., Roche–Newton, O. and Senger, S.. Convexity, superquadratic growth, and dot products. J. London Math. Soc. 107(5) (2023), 1900–1923.CrossRefGoogle Scholar
Roche–Newton, O. and Zhelezov, D.. Convexity, elementary methods, and distances. To appear in Discrete Comput. Geom. Google Scholar
Rudnev, M. and Stevens, S.. An update on the sum-product problem. Math. Proc. Camb. Phil. Soc. 173(2) (2022), 411–430.CrossRefGoogle Scholar
Ruzsa, I. Z., Shakan, G., Solymosi, J. and Szemerédi, E.. On distinct consecutive differences. Combinatorial and additive number theory IV (Springer Proc. Math. Stat., 347, Springer, Cham, 2021), 425–434.CrossRefGoogle Scholar
Zhelezov, D. and Pálvőlgyi, D.. Query complexity and the polynomial Freiman–Ruzsa conjecture. Adv. Math. 392 (2021), 402408.CrossRefGoogle Scholar