Stronger uniform approximate orthogonality for nilspace characters

Prove the stronger uniform estimate $\sup_{h\in \ab}\|F_\chi\circ\phi\,\overline{T^hF_{\chi'}\circ\phi}\|_{U^k}$ is small for distinct vertical frequencies $\chi\neq\chi'$, rather than only establishing separate smallness of the ordinary inner product and the $U^{k+1}$-product.

Background

The paper establishes two approximate orthogonality statements for distinct nilspace-character components: small ordinary inner product and small Uk+1U^{k+1}-product. It observes that a single stronger translation-uniform estimate would imply both results, but does not prove it and suggests that a finer analysis of the underlying nilspace structures may be needed.

References

We strongly believe that the two main results of this subsection are in fact consequences of a single stronger result, which would constitute in various ways a more natural form of approximate orthogonality. More precisely, we believe that, rather than the smallness of $\langle F_\chi\circ\phi,F_{\chi'}\circ\phi\rangle_{U{k+1}$ and of $\langle F_\chi\circ\phi,F_{\chi'}\circ\phi\rangle$ (established in these results), what holds is the following property, which implies the previous two: the smallness of $\sup_{h\in \ab}|F_\chi\circ\phi \,\overline{ThF_{\chi'}\circ\phi}|_{Uk}$.

— Spectral algorithms in higher-order Fourier analysis  (2501.12287 - Candela et al., 21 Jan 2025) in Remark following Theorem 5.3, Section 5.2