ROTH’S THEOREM FOR FOUR VARIABLES AND ADDITIVE STRUCTURES IN SUMS OF SPARSE SETS
Forum of Mathematics, Sigma, Tome 4 (2016)
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We show that if $A\subset \{1,\ldots ,N\}$ does not contain any nontrivial solutions to the equation $x+y+z=3w$ , then
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.
| $\begin{eqnarray}|A|\leqslant \frac{N}{\exp (c(\log N)^{1/7})},\end{eqnarray}$ |
@article{10_1017_fms_2016_2,
author = {TOMASZ SCHOEN and OLOF SISASK},
title = {ROTH{\textquoteright}S {THEOREM} {FOR} {FOUR} {VARIABLES} {AND} {ADDITIVE} {STRUCTURES} {IN} {SUMS} {OF} {SPARSE} {SETS}},
journal = {Forum of Mathematics, Sigma},
publisher = {mathdoc},
volume = {4},
year = {2016},
doi = {10.1017/fms.2016.2},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2016.2/}
}
TY - JOUR AU - TOMASZ SCHOEN AU - OLOF SISASK TI - ROTH’S THEOREM FOR FOUR VARIABLES AND ADDITIVE STRUCTURES IN SUMS OF SPARSE SETS JO - Forum of Mathematics, Sigma PY - 2016 VL - 4 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.1017/fms.2016.2/ DO - 10.1017/fms.2016.2 LA - en ID - 10_1017_fms_2016_2 ER -
TOMASZ SCHOEN; OLOF SISASK. ROTH’S THEOREM FOR FOUR VARIABLES AND ADDITIVE STRUCTURES IN SUMS OF SPARSE SETS. Forum of Mathematics, Sigma, Tome 4 (2016). doi: 10.1017/fms.2016.2
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