Estimates for $L^p$ variants of Gowers norms

Vjekoslav Kovač; Keith M. Rogers

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arXiv · 2609.08916

Estimates for $L^p$ variants of Gowers norms

Vjekoslav Kovač·Keith M. Rogers

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Abstract

Motivated by a log-convexity question of Bennett and Tao, we consider Gowers-type functionals defined by $L^p$ norms of multiple autocorrelations. We prove sharp bounds in terms of $L^q$ norms and characterise the near-extremisers on Euclidean spaces as well as locally compact abelian groups. We also establish two families of degree-lowering inequalities and study their near-extremisers. As a byproduct of this broader theory, we show that the constant in the log-convexity estimate for Gowers norms is strictly less than unity, confirming the aforementioned conjecture of Bennett and Tao. Finally, we characterise the values of the parameters for which these Gowers-type functionals necessarily satisfy the triangle inequality on nonnegative measurable functions.

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BibTeX RIS

Vjekoslav Kovač, Keith M. Rogers. 2026-09-08. Estimates for $L^p$ variants of Gowers norms. https://arxiv.org/abs/2609.08916

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