Oliver Roche-Newton

Variations on the sum-product problem

This paper is a sequel to a paper entitled Variations on the sum-product problem by the same authors [SIAM J. Discrete Math., 29 (2015), pp. 514-540]. In this sequel, we quantitatively improve several of the main results of the first paper as well as generalize a method from it to give a near-optimal bound for a new expander. The main new results are the following bounds, which hold for any finite set

Sets with few distinct distances do not have heavy lines

A \subset \mathbb R

An improved sum-product estimate for general finite fields

:

On distinct perpendicular bisectors and pinned distances in finite fields

\exists a \in A

New Sum-Product Estimates for Real and Complex Numbers

such that

On the size of the set AA+A: ON THE SIZE OF THE SET AA+A

|A(A+a)| \gtrsim |A|^{\frac{3}{2}+\frac{1}{186}}, |A(A-A)| \gtrsim |A|^{\frac{3}{2}+\frac{1}{34}}, |A(A+A)| \gtrsim |A|^{\frac{3}{2}+\frac{5}{242}}, |\{(a_1+a_2+a_3+a_4)^2+\log a_5 : a_i \in A \}| \gg \frac{|A|^2}{\log |A|}

New sum-product type estimates over finite fields

.

On an Application of Guth–Katz Theorem

Let P be a set of n points in the plane that determines at most n / 5 distinct distances. We show that no line can contain more than O ( n 43 / 52 polylog ( n ) ) points of P . We also show a similar result for rectangular distances, equivalent to distances in the Minkowski plane, where the distance between a pair of points is the area of the axis-parallel rectangle that they span.

New results on sum-product type growth over fields

An improved sum-product estimate for subsets of a finite field whose order is not prime is provided. It is shown, under certain conditions, that max{|A+A|,|A·A|}≫|A|12/11(log2|A|)5/11. This new estimate matches, up to a logarithmic factor, the current best known bound obtained over prime fields by Rudnev.

A Short Proof of a Near-Optimal Cardinality Estimate for the Product of a Sum Set

Given a set of points