Sharp asymmetric off-diagonal Young region beyond the symmetric endpoint

Determine the larger sharp asymmetric off-diagonal exponent region for varieties with intrinsic exponent greater than two by establishing genuinely mixed geometric energy estimates beyond those obtained from Cauchy–Schwarz applied to two self-energy bounds.

Background

The paper derives off-diagonal convolution inequalities by combining self-energy estimates for two finite sets and interpolating the resulting symmetric Fourier-extension endpoint with the classical Young endpoints. When the intrinsic exponent is greater than two, this yields an interpolation polygon, but the authors identify a possible larger sharp region that cannot be obtained from self-energy information alone.

The unresolved issue is to determine whether mixed geometric energy estimates can enlarge this region and, if so, to characterize the optimal asymmetric exponents. This question is distinct from the near-diagonal case, for which the paper states a sharp region.

References

A second open direction concerns the off-diagonal region when \alpha(V)>2. Self-energy supplies a symmetric \ell{4/\alpha}\to L4 endpoint, and \cref{thm:general-offdiag} gives its interpolation polygon. A larger sharp asymmetric region would require genuinely mixed geometric energy estimates rather than Cauchy--Schwarz between two self-energies.

— Complexity-sensitive additive energy and off-diagonal Young inequalities on bounded-degree algebraic varieties  (2608.18956 - Hu, 19 Aug 2026) in Section 6, “Higher-dimensional consequences, sharpness, and limitations,” subsection “What the flag theorem does not remove”