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Estimates for Lp 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 Lp norms of multiple autocorrelations. We prove sharp bounds in terms of Lq 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 p for which the Gowers-type functionals necessarily satisfy the triangle inequality on nonnegative measurable functions.

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Cite

BibTeXRIS
BibTeX
@misc{kovac2026estimates,
  title = {{Estimates for $L^p$ variants of Gowers norms}},
  author = {Vjekoslav Kovač and Keith M. Rogers},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.08916},
  eprint = {2609.08916},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.08916},
}
RIS
TY  - UNPB
TI  - Estimates for L^p variants of Gowers norms
AU  - Vjekoslav Kovač
AU  - Keith M. Rogers
PY  - 2026
DA  - 2026/10/14
PB  - arXiv
DO  - 10.48550/arxiv.2609.08916
UR  - https://arxiv.org/abs/2609.08916
AB  - 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 p for which the Gowers-type functionals necessarily satisfy the triangle inequality on nonnegative measurable functions.
ER  -