Bennett–Tao log-convexity conjecture for Gowers norms

The Bennett–Tao conjecture asks for a uniform strengthening of log-convexity inequalities between Gowers norms, with analogous questions for higher uniformities.

Known results

• For measurable $f:\mathbb{R}\to\mathbb{R}^{+}$, earlier work records the conjectured bound $\|f\|_{U^{2}}\le(1-\varepsilon)\|f\|_{U^{1}}^{1/2}\|f\|_{U^{3}}^{1/2}$ for some $\varepsilon>0$, but does not establish it.
• It proves only strict non-equality: $\|f\|_{U^{2}}<\|f\|_{U^{1}}^{1/2}\|f\|_{U^{3}}^{1/2}$ when the relevant norms are finite and nonzero.
• Sharp individual inequalities for the $U^{d}$ norms do not imply the conjectured uniform gap.

September 2026 claimed resolution

A September 2026 arXiv preprint, Estimates for $L^{p}$ variants of Gowers norms, claims broad sharp inequalities and confirmation of the Bennett–Tao conjecture. This is an unrefereed claim and has not been independently verified.

Current status (as of September 2026): The conjecture is claimed solved by an unrefereed preprint, but remains unverified; earlier literature established only strict non-equality, not the conjectured uniform gap.