Average-Value Tverberg Partitions via Finite Fourier Analysis

TL;DR

This paper explores average-value coincidences in Tverberg partitions for continuous maps, using finite Fourier analysis to reveal new intersection properties even when the classical conjecture fails.

Contribution

It introduces a novel Fourier-analytic approach to Tverberg-type problems, establishing average-value intersection results below the classical dimension threshold.

Findings

Average-value coincidences hold for continuous maps below the Tverberg dimension.

Fourier transforms' vanishing characterizes these average-value intersections.

Results extend to non-prime-power cases where the classical conjecture fails.

Abstract

The long-standing topological Tverberg conjecture claimed, for any continuous map from the boundary of an $N(q,d):=(q-1)(d+1)$-simplex to $d$-dimensional Euclidian space, the existence of $q$ pairwise disjoint subfaces whose images have non-empty $q$-fold intersection. The affine cases, true for all $q$, constitute Tverberg's famous 1966 generalization of the classical Radon's Theorem. Although established for all prime powers in 1987 by \"Ozaydin, counterexamples to the conjecture, relying on 2014 work of Mabillard and Wagner, were first shown to exist for all non-prime-powers in 2015 by Frick. Starting with a reformulation of the topological Tverberg conjecture in terms of harmonic analysis on finite groups, we show that despite the failure of the conjecture, continuous maps \textit{below} the tight dimension $N(q,d)$ are nonetheless guaranteed $q$ pairwise disjoint subfaces -- including when $q$ is not a prime power -- which satisfy a variety of "average value" coincidences, the latter obtained as the vanishing of prescribed Fourier transforms.