  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.
