On sumsets of convex sets

A set of reals A={a_1,...,a_2} is called convex if a_{i+1} - a_i > a_i - a_{i-1} for all i. We prove, in particular, that |A-A| \gg |A|^{8/5} \log{-2/5} |A|.Comment: 6 page

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

Improved bounds on the set A(A+1)

For a subset A of a field F, write A(A + 1) for the set {a(b + 1):a,b\in A}. We establish new estimates on the size of A(A+1) in the case where F is either a finite field of prime order, or the real line. In the finite field case we show that A(A+1) is of cardinality at least C|A|^{57/56-o(1)} for some absolute constant C, so long as |A| < p^{1/2}. In the real case we show that the cardinality is at least C|A|^{24/19-o(1)}. These improve on the previously best-known exponents of 106/105-o(1) and 5/4 respectively

Variations on the sum-product problem II

This is a sequel to the paper arXiv:1312.6438 by the same authors. In this sequel, we quantitatively improve several of the main results of arXiv:1312.6438, and build on the methods therein. The main new results is that, for any finite set $A \subset \mathbb R$, there exists $a \in A$ such that $|A(A+a)| \gtrsim |A|^{\frac{3}{2}+\frac{1}{186}}$. We give improved bounds for the cardinalities of $A(A+A)$ and $A(A-A)$. Also, we prove that $|\{(a_1+a_2+a_3+a_4)^2+\log a_5 : a_i \in A \}| \gg \frac{|A|^2}{\log |A|}$. The latter result is optimal up to the logarithmic factor.Comment: This paper supersedes arXiv:1603.0682

On sums of Szemer\'edi--Trotter sets

We prove new general results on sumsets of sets having Szemer\'edi--Trotter type. This family includes convex sets, sets with small multiplicative doubling, images of sets under convex/concave maps and others.Comment: 12 page