# A note on norms of signed sums of vectors

## TL;DR

This paper improves bounds on the norms of signed sums of vectors, showing that for certain large sets of sign patterns, one can find vectors with large infinity norm sums, extending results to random points and arbitrary norms.

## Contribution

It provides new bounds on the norms of signed sums of vectors for large sets, generalizing previous results and including random vectors and arbitrary norms.

## Key findings

- Existence of vectors with large signed sum norms for sets of size up to 2^{n/f(n)}
- Extension of results to random vectors uniformly distributed in convex bodies
- Generalization to arbitrary norms on R^n"

## 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 ${\mathbb R}^n$.