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

TOMASZ SCHOEN, OLOF SISASK
2016 Forum of Mathematics, Sigma  
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
more » ... $, or high-dimensional affine subspaces if$A\subset \mathbb{F}_{q}^{n}$, even if$A$has density of the shape above.
doi:10.1017/fms.2016.2 fatcat:j4itpqbqdjhzbiqrqptbs6csqu