A root-system reduction for the spherical triangle form

Definitions and mathematical statement

Let $S^n=\{x\in\mathbb R^{n+1}:\|x\|_2=1\}$, $n\geq1$, with normalized rotation-invariant measure $\sigma_n$. Choose any fixed ordered triple $x_0,y_0,z_0\in S^n$ satisfying $x_0+y_0+z_0=0$. Its three pairwise inner products are $-1/2$. For a measurable real function $f$ with $|f|\leq1$, define \[ T(f)=\mathbb E_R f(Rx_0)f(Ry_0)f(Rz_0), \] where $R$ is Haar-uniform on the orthogonal group $O(n+1)$. This definition specifies the invariant probability measure on the constrained triples and is independent of the initially chosen triple.

The proposed inverse theorem asks whether for every $\delta\in(0,1)$ there exist $m(\delta)\in\mathbb N$ and $c(\delta)>0$, independent of $n$, with the following property. Whenever $|T(f)|>\delta$, there are an orthonormal family $u_1,\ldots,u_s$ with $s\leq\min\{m(\delta),n+1\}$ and a measurable function $g:\mathbb R^s\to[-1,1]$ such that \[ \left|\int_{S^n}f(x)\, g(\langle x,u_1\rangle,\ldots,\langle x,u_s\rangle) \,d\sigma_n(x)\right|\geq c(\delta). \] Thus both the number of coordinates and the correlation threshold are uniform in ambient dimension. The coordinates may be chosen after seeing $f$; they need not be the original coordinate axes. The bound on $g$ fixes the normalization of "correlation" and prevents rescaling a negligible correlation into a meaningful one.

Short English statement

Must a bounded function with a large equilateral-triangle average correlate with a function of only boundedly many suitably chosen coordinates?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20000882 (AIM-COMBINATORICS-0007), AIM Workshop Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 American Institute of Mathematics, High-dimensional phenomena in discrete analysis, item 2.4. Original workshop question underlying AIM-COMBINATORICS-0007. http://aimpl.org/highdimdiscrete/2/.