ROTH’S THEOREM FOR FOUR VARIABLES AND ADDITIVE STRUCTURES IN SUMS OF SPARSE SETS

Abstract

We show that if $A\subset \{1,\ldots ,N\}$ does not contain any nontrivial solutions to the equation $x+y+z=3w$, then $$\begin{eqnarray}|A|\leqslant \frac{N}{\exp (c(\log N)^{1/7})},\end{eqnarray}$$ where $c>0$ is some absolute constant. In view of Behrend’s construction, this bound is of the right shape: the exponent $1/7$ cannot be replaced by any constant larger than $1/2$. We also establish a related result, which says that sumsets $A+A+A$ contain long arithmetic progressions if $A\subset \{1,\ldots ,N\}$, or high-dimensional affine subspaces if $A\subset \mathbb{F}_{q}^{n}$, even if $A$ has density of the shape above.