Vector Balancing in Lebesgue Spaces

A tantalizing conjecture in discrete mathematics is the one of Koml\'os, suggesting that for any vectors $\mathbf{a}1,\ldots,\mathbf{a}n \in B2^m$ there exist signs $x1, \dots, xn \in { -1,1}$ so that $|\sum{i=1}ⁿ xi\mathbf{a}i|\infty \le O(1)$. It is a natural extension to ask what $\ellq$-norm bound to expect for $\mathbf{a}1,\ldots,\mathbf{a}n \in Bp^m$. We prove that, for $2 \le p \le q \le \infty$, such vectors admit fractional colorings $x1, \dots, xn \in [-1,1]$ with a linear number of $\pm 1$ coordinates so that $|\sum{i=1}ⁿ xi\mathbf{a}i|q \leq O(\sqrt{\min(p,\log(2m/n))}) \cdot n^{1/2-1/p+ 1/q}$, and that one can obtain a full coloring at the expense of another factor of $\frac{1}{1/2 - 1/p + 1/q}$. In particular, for $p \in (2,3]$ we can indeed find signs $\mathbf{x} \in { -1,1}^n$ with $|\sum{i=1}ⁿ xi\mathbf{a}i|_\infty \le O(n^{1/2-1/p} \cdot \frac{1}{p-2})$. Our result generalizes Spencer's theorem, for which $p = q = \infty$, and is tight for $m = n$. Additionally, we prove that for any fixed constant $\delta>0$, in a centrally symmetric body $K \subseteq \mathbb{R}^n$ with measure at least $e^{-\delta n}$ one can find such a fractional coloring in polynomial time. Previously this was known only for a small enough constant -- indeed in this regime classical nonconstructive arguments do not apply and partial colorings of the form $\mathbf{x} \in { -1,0,1}^n$ do not necessarily exist.