A note on norms of signed sums of vectors

Abstract

Our starting point is an improved version of a result of D. Hajela related to a question of Koml\'{o}s: we show that if $f(n)$ is a function such that $\lim\limits_{n\to\infty }f(n)=\infty $ and $f(n)=o(n)$, there exists $n_0=n_0(f)$ such that for every $n\geqslant n_0$ and any $S\subseteq \{-1,1\}^n$ with cardinality $|S|\leqslant 2^{n/f(n)}$ one can find orthonormal vectors $x_1,\ldots ,x_n\in {\mathbb R}^n$ that satisfy $$\|\epsilon_1x_1+\cdots +\epsilon_nx_n\|_{\infty }\geqslant c\sqrt{\log f(n)}$$ for all $(\epsilon_1,\ldots ,\epsilon_n)\in S$. We obtain analogous results in the case where $x_1,\ldots ,x_n$ are independent random points uniformly distributed in the Euclidean unit ball $B_2^n$ or any symmetric convex body, and the $\ell_{\infty }^n$-norm is replaced by an arbitrary norm on ${\mathbb R}^n$.