Non-degenerate Hilbert cubes in random sets

@article{Sándor2007,
abstract = {A slight modification of the proof of Szemerédi’s cube lemma gives that if a set $S\subset [1,n]$ satisfies $|S|\ge \frac\{n\}\{2\}$, then $S$ must contain a non-degenerate Hilbert cube of dimension $\lfloor \log _2\log _2n -3\rfloor $. In this paper we prove that in a random set $S$ determined by $\textrm\{Pr\}\lbrace s\in S\rbrace =\frac\{1\}\{2\}$ for $1\le s\le n$, the maximal dimension of non-degenerate Hilbert cubes is a.e. nearly $\log _2\log _2n+\log _2\log _2\log _2n$ and determine the threshold function for a non-degenerate $k$-cube.},
affiliation = {Institute of Mathematics Budapest University of Technology and Economics Egry J. u. 1., H-1111 Budapest, Hungary},
author = {Sándor, Csaba},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Hilbert cube; subset sum; random set},
language = {eng},
number = {1},
pages = {249-261},
publisher = {Université Bordeaux 1},
title = {Non-degenerate Hilbert cubes in random sets},
url = {http://eudml.org/doc/249953},
volume = {19},
year = {2007},
}

TY - JOUR
AU - Sándor, Csaba
TI - Non-degenerate Hilbert cubes in random sets
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2007
PB - Université Bordeaux 1
VL - 19
IS - 1
SP - 249
EP - 261
AB - A slight modification of the proof of Szemerédi’s cube lemma gives that if a set $S\subset [1,n]$ satisfies $|S|\ge \frac{n}{2}$, then $S$ must contain a non-degenerate Hilbert cube of dimension $\lfloor \log _2\log _2n -3\rfloor $. In this paper we prove that in a random set $S$ determined by $\textrm{Pr}\lbrace s\in S\rbrace =\frac{1}{2}$ for $1\le s\le n$, the maximal dimension of non-degenerate Hilbert cubes is a.e. nearly $\log _2\log _2n+\log _2\log _2\log _2n$ and determine the threshold function for a non-degenerate $k$-cube.
LA - eng
KW - Hilbert cube; subset sum; random set
UR - http://eudml.org/doc/249953
ER -