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 Tverbergs famous 1966 generalization of the classical Radons 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.
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