A counterexample to the Gowers $U^k$ uncertainty conjecture for $k\ge 6$

Abstract

The Donoho--Stark uncertainty principle asserts $N^d\le |E||\Sigma |$, and the additive energy uncertainty principle of Aldahleh--Iosevich--Iosevich--Jaimangal--Mayeli--Pack refines it to $1\le |E|\cdot \Vert 1_{\Sigma }{\Vert }_{U^2}^{4/3}$. It has been conjectured that the natural higher-order analogue \[ 1\le |E|\cdot\|1_Σ\|_{U^k}^{2^k/(k+1)} \] holds for every $k\ge 2$, the coset case giving equality. We show that this fails for every $k\ge 6$. The counterexample is the two-point signal $f={\delta }_0-{\delta }_1$ on ${\mathbb{Z}}_3$, for which the left-hand quantity equals $2/3(2(k+1){)}^{1/(k+1)}$; this is $<1$ precisely when $k\ge 6$, and tends to $2/3$.