A short proof of a near-optimal cardinality estimate for the product of a sum set

In this note it is established that, for any finite set $A$ of real numbers, there exist two elements $a,b \in A$ such that $$|(a+A)(b+A)| \gg \frac{|A|^2}{\log |A|}.$$ In particular, it follows that $|(A+A)(A+A)| \gg \frac{|A|^2}{\log |A|}$. The latter inequality had in fact already been established in an earlier work of the author and Rudnev (arXiv:1203.6237), which built upon the recent developments of Guth and Katz (arXiv:1011.4105) in their work on the Erd\H{o}s distinct distance problem. Here, we do not use those relatively deep methods, and instead we need just a single application of the Szemer\'{e}di-Trotter Theorem. The result is also qualitatively stronger than the corresponding sum-product estimate from (arXiv:1203.6237), since the set $(a+A)(b+A)$ is defined by only two variables, rather than four. One can view this as a solution for the pinned distance problem, under an alternative notion of distance, in the special case when the point set is a direct product $A \times A$. Another advantage of this more elementary approach is that these results can now be extended for the first time to the case when $A \subset \mathbb C$.Comment: To appear in Proceedings of SoCG 201

Convexity and a sum-product type estimate

In this paper we further study the relationship between convexity and additive growth, building on the work of Schoen and Shkredov (\cite{SS}) to get some improvements to earlier results of Elekes, Nathanson and Ruzsa (\cite{ENR}). In particular, we show that for any finite set $A\subset{\mathbb{R}}$ and any strictly convex or concave function $f$, $$|A+f(A)|\gg{\frac{|A|^{24/19}}{(\log|A|)^{2/19}}}$$ and $$\max\{|A-A|,\ |f(A)+f(A)|\}\gg{\frac{|A|^{14/11}}{(\log|A|)^{2/11}}}.$$ For the latter of these inequalities, we go on to consider the consequences for a sum-product type problem

Four-term progression free sets with three-term progressions in all large subsets

This paper is mainly concerned with sets which do not contain four-term arithmetic progressions, but are still very rich in three term arithmetic progressions, in the sense that all sufficiently large subsets contain at least one such progression. We prove that there exists a positive constant $c$ and a set $A \subset \mathbb F_q^n$ which does not contain a four-term arithmetic progression, with the property that for every subset $A' \subset A$ with $|A'| \geq |A|^{1-c}$, $A'$ contains a nontrivial three term arithmetic progression. We derive this from a more general quantitative Roth-type theorem in random subsets of $\mathbb{F}_{q}^{n}$, which improves a result of Kohayakawa-Luczak-R\"odl/Tao-Vu. We also discuss a similar phenomenon over the integers, where we show that for all $\epsilon >0$, and all sufficiently large $N \in \mathbb N$, there exists a four-term progression-free set $A$ of size $N$ with the property that for every subset $A' \subset A$ with $|A'| \gg \frac{1}{(\log N)^{1-\epsilon}} \cdot N$ contains a nontrivial three term arithmetic progression. Finally, we include another application of our methods, showing that for sets in $\mathbb{F}_{q}^{n}$ or $\mathbb{Z}$ the property of "having nontrivial three-term progressions in all large subsets" is almost entirely uncorrelated with the property of "having large additive energy".Comment: minor updates including suggestions from referee