Let $v_1$, $v_2$, ..., $v_n$ be real numbers whose squares add up to 1. Consider the $2^n$ signed sums of the form $S = \sum \pm v_i$. Holzman and Kleitman (1992) proved that at least 3/8 of these sums satisfy $|S| \le 1$. This 3/8 bound seems to be the best their method can achieve. Using a different method, we improve the bound to 13/32, thus breaking the 3/8 barrier.
@article{10_37236_6949,
author = {Ravi B. Boppana and Ron Holzman},
title = {Tomaszewski's problem on randomly signed sums: breaking the 3/8 barrier},
journal = {The electronic journal of combinatorics},
year = {2017},
volume = {24},
number = {3},
doi = {10.37236/6949},
zbl = {1381.60025},
url = {http://geodesic.mathdoc.fr/articles/10.37236/6949/}
}
TY - JOUR
AU - Ravi B. Boppana
AU - Ron Holzman
TI - Tomaszewski's problem on randomly signed sums: breaking the 3/8 barrier
JO - The electronic journal of combinatorics
PY - 2017
VL - 24
IS - 3
UR - http://geodesic.mathdoc.fr/articles/10.37236/6949/
DO - 10.37236/6949
ID - 10_37236_6949
ER -
%0 Journal Article
%A Ravi B. Boppana
%A Ron Holzman
%T Tomaszewski's problem on randomly signed sums: breaking the 3/8 barrier
%J The electronic journal of combinatorics
%D 2017
%V 24
%N 3
%U http://geodesic.mathdoc.fr/articles/10.37236/6949/
%R 10.37236/6949
%F 10_37236_6949
Ravi B. Boppana; Ron Holzman. Tomaszewski's problem on randomly signed sums: breaking the 3/8 barrier. The electronic journal of combinatorics, Tome 24 (2017) no. 3. doi: 10.37236/6949
