Divergence of decreasing rearranged Fourier series on every infinite compact abelian group

Abstract

On every infinite compact Hausdorff abelian group there is a complex-valued continuous function whose Fourier sums, taken over coefficients above a decreasing magnitude threshold, are unbounded almost everywhere. In fact, such functions form a dense \(G_\delta\) in the space of continuous functions with its uniform norm. Moreover, a divergent example may be chosen with Fourier support contained in any prescribed infinite subgroup of the dual, while avoiding any prescribed finite set of frequencies. No metrizability or zero-dimensionality is assumed. The proof combines K\"orner's finite circle and Walsh lemmas with a finite construction on products of odd cyclic groups. A trichotomy for the dual group, exact quotient transfer, and cyclic sampling give finite examples on every group under consideration.