Jackson's inequality on the hypercube

TL;DR

This paper explores bounds on Jackson's inequality related to functions on the hypercube, revealing limitations of sensitivity-based bounds, and establishing new results on approximate degree, symmetry, and subspace approximation.

Contribution

It provides new lower bounds for Jackson's inequality on the hypercube, demonstrates the failure of the sensitivity theorem for bounded functions, and introduces bounds for symmetric functions and subspace approximation.

Findings

J(n, 0.499n) is bounded below by a positive constant independent of n

Reverse Bernstein inequality fails in the tail space for certain parameters

Existence of functions with constant sensitivity but linearly growing approximate degree

Abstract

We investigate the best constant $J(n,d)$ such that Jackson's inequality \[ \inf_{\mathrm{deg}(g) \leq d} \|f - g\|_{\infty} \leq J(n,d) \, s(f), \] holds for all functions $f$ on the hypercube $\{0,1\}^n$, where $s(f)$ denotes the sensitivity of $f$. We show that the quantity $J(n, 0.499n)$ is bounded below by an absolute positive constant, independent of $n$. This complements Wagner's theorem, which establishes that $J(n,d)\leq 1 $. As a first application we show that reverse Bernstein inequality fails in the tail space $L^{1}_{\geq 0.499n}$ improving over previously known counterexamples in $L^{1}_{\geq C \log \log (n)}$. As a second application, we show that there exists a function $f : \{0,1\}^n \to [-1,1]$ whose sensitivity $s(f)$ remains constant, independent of $n$, while the approximate degree grows linearly with $n$. This result implies that the sensitivity theorem $s(f) \geq \Omega(\mathrm{deg}(f)^C)$ fails in the strongest sense for bounded real-valued functions even when $\mathrm{deg}(f)$ is relaxed to the approximate degree. We also show that in the regime $d = (1 - \delta)n$, the bound \[ J(n,d) \leq C \min\{\delta, \max\{\delta^2, n^{-2/3}\}\} \] holds. Moreover, when restricted to symmetric real-valued functions, we obtain $J_{\mathrm{symmetric}}(n,d) \leq C/d$ and the decay $1/d$ is sharp. Finally, we present results for a subspace approximation problem: we show that there exists a subspace $E$ of dimension $2^{n-1}$ such that $\inf_{g \in E} \|f - g\|_{\infty} \leq s(f)/n$ holds for all $f$.