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A better than $3/2$ exponent for iterated sums and products over $\mathbb R
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...
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Published in: | Mathematical proceedings of the Cambridge Philosophical Society 2024-07, Vol.177 (1), p.11-22 |
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Main Author: | |
Format: | Article |
Language: | English |
Citations: | Items that this one cites |
Online Access: | Get full text |
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Summary: | 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. |
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ISSN: | 0305-0041 1469-8064 |
DOI: | 10.1017/S0305004124000112 |