Balancing Polynomials in the Chebyshev Norm

TL;DR

This paper proves that for any set of bounded degree polynomials, one can assign signs to keep their sum uniformly small in the Chebyshev norm, extending classical results and connecting to Spencer's theorem.

Contribution

It introduces a new bound for balancing polynomials in the Chebyshev norm, generalizing the Rudin-Shapiro sequence and polynomial analogues of Spencer's theorem.

Findings

Existence of signs with Chebyshev norm less than 30√n

Extension of Rudin-Shapiro sequence bounds

Connection to Spencer's six standard deviations theorem

Abstract

Given $n$ polynomials $p_1, \dots, p_n$ of degree at most $n$ with $\|p_i\|_\infty \le 1$ for $i \in [n]$, we show there exist signs $x_1, \dots, x_n \in \{-1,1\}$ so that \[\Big\|\sum_{i=1}^n x_i p_i\Big\|_\infty < 30\sqrt{n}, \] where $\|p\|_\infty := \sup_{|x| \le 1} |p(x)|$. This result extends the Rudin-Shapiro sequence, which gives an upper bound of $O(\sqrt{n})$ for the Chebyshev polynomials $T_1, \dots, T_n$, and can be seen as a polynomial analogue of Spencer's "six standard deviations" theorem.