Brendan Murphy

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

Growth Estimates in Positive Characteristic via Collisions

A \subset \mathbb R

New results on sum-product type growth over fields

:

A point-line incidence identity in finite fields, and applications

\exists a \in A

Popular Products and Continued Fractions

such that

Products of Differences over Arbitrary Finite Fields

|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|}

On the few products, many sums problem

.