# A new expander and improved bounds for $A(A+A)$

Research paper by **Oliver Roche-Newton**

Indexed on: **22 Mar '16**Published on: **22 Mar '16**Published in: **Mathematics - Combinatorics**

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#### Abstract

The main result in this paper concerns a new five-variable expander. It is
proven that for any finite set of real numbers $A$,
$$|\{(a_1+a_2+a_3+a_4)^2+\log a_5 :a_1,a_2,a_3,a_4,a_5 \in A \}| \gg
\frac{|A|^2}{\log |A|}.$$ This bound is optimal, up to logarithmic factors. The
paper also gives new lower bounds for $|A(A-A)|$ and $|A(A+A)|$, improving on
results from arXiv:1312.6438. The new bounds are $$|A(A-A)| \gtrapprox
|A|^{3/2+\frac{1}{34}}$$ and $$|A(A+A)| \gtrapprox |A|^{3/2+\frac{5}{242}}.$$