Using recent developments on the theory of locally decodable codes, we prove that the critical size for Szemerédi's theorem with random differences is bounded from above by $N^{1-\frac{2}{k} + o(1)}$ for length-$k$ progressions. This improves the previous best bounds of $N^{1-\frac{1}{\lceil k/2 \rceil} + o(1)}$ for all odd $k$.
@article{10_37236_12415,
author = {Jop Bri\"et and Davi Castro-Silva},
title = {On the threshold for {Szemer\'edi's} theorem with random differences},
journal = {The electronic journal of combinatorics},
year = {2024},
volume = {31},
number = {4},
doi = {10.37236/12415},
zbl = {1554.11008},
url = {http://geodesic.mathdoc.fr/articles/10.37236/12415/}
}
TY - JOUR
AU - Jop Briët
AU - Davi Castro-Silva
TI - On the threshold for Szemerédi's theorem with random differences
JO - The electronic journal of combinatorics
PY - 2024
VL - 31
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/12415/
DO - 10.37236/12415
ID - 10_37236_12415
ER -
%0 Journal Article
%A Jop Briët
%A Davi Castro-Silva
%T On the threshold for Szemerédi's theorem with random differences
%J The electronic journal of combinatorics
%D 2024
%V 31
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/12415/
%R 10.37236/12415
%F 10_37236_12415
Jop Briët; Davi Castro-Silva. On the threshold for Szemerédi's theorem with random differences. The electronic journal of combinatorics, Tome 31 (2024) no. 4. doi: 10.37236/12415
