Measure Equipartitions via Finite Fourier Analysis

Applications of harmonic analysis on finite groups are introduced to measure partition problems, with equipartitions obtained as the vanishing of prescribed Fourier transforms. For elementary abelian groups $Z_p^k$, $p$ an odd prime, equipartitions are by $k$-tuples of complex regular $p$-fans in $\mathbb{C}^d$, analogues of the famous Gr\"unbaum problem on equipartitions in $\mathbb{R}^d$ by $k$-tuples of hyperplanes (i.e., regular 2-fans). Here the number of regions is a prime power, as usual in topological applications to combinatorial geometry. For general abelian groups, however, the Fourier perspective yields new classes of equipartitions by families of complex regular fans $F_{q_1},\ldots, F_{q_k}$ (such as those of a "Makeev-type"), including when the number of regions is not a prime power.