Recurrence and non-uniformity of bracket polynomials

In his celebrated proof of Szemerédi’s theorem that a set of integers of positive density contains arbitrarily long arithmetic progressions, W. T. Gowers introduced a certain sequence of norms \( \|\cdot\|_{U^2[\mathbb{N}]} \leq \|\cdot\|_{U^3[\mathbb{N}]} \leq \cdots \) on the space of complex-valued functions on the set \( [N] \). An important question regarding these norms concerns for which functions they are `large’ in a certain sense.

This question has been answered fairly completely by B. Green, T. Tao and T. Ziegler in terms of certain algebraic functions called \textit{nilsequences}. In this work, we show that more explicit functions called \textit{bracket polynomials} have `large’ Gowers norm. Specifically, for a fairly large class of bracket polynomials, called \textit{constant-free bracket polynomials}, we show that if \( \phi \) is a bracket polynomial of degree \( k-1 \) on \( [N] \), then the function \( f : n \mapsto e(\phi(n)) \) has Gowers \( U^k[\mathbb{N}] \)-norm uniformly bounded away from zero.

We establish this result by first reducing it to a certain recurrence property of sets of constant-free bracket polynomials. Specifically, we show that if \( \theta_1, \ldots, \theta_r \) are constant-free bracket polynomials, then their values, modulo 1, are all close to zero on at least some constant proportion of the points \( 1, \ldots, N \).

The proof of this statement relies on two deep results from the literature. The first is work of V. Bergelson and A. Leibman showing that an arbitrary bracket polynomial can be expressed in terms of a so-called \textit{polynomial sequence} on a nilmanifold. The second is a theorem of B. Green and T. Tao describing the quantitative distribution properties of such polynomial sequences.

In the special cases of the bracket polynomials \( \phi_{k-1}(n) = \alpha_{k-1} n^{k-2} \{ \alpha_1 n \cdots \} \) with \( k \leq 5 \), we give elementary alternative proofs of the fact that \( \|\phi_{k-1}\|_{U^k[\mathbb{N}]} \) is `large,’ without reference to nilmanifolds. Here we write \( \{x\} \) for the fractional part of \( x \), chosen to lie in \( (-1/2, 1/2] \).