Exponential sums and Riesz energies

We bound an exponential sum that appears in the study of irregularities of distribution (the low-frequency Fourier energy of the sum of several Dirac measures) by geometric quantities: a special case is that for all {x1,…,xN}⊂T2, X≥1 and a universal c>0∑i,j=1NX21+X4‖xi−xj‖4≲∑k∈Z2‖k‖≤X|∑n=1Ne2πi〈k,xn〉|2≲∑i,j=1NX2e−cX2‖xi−xj‖2. Since this exponential sum is intimately tied to rather subtle distribution properties of the points, we obtain nonlocal structural statements for near-minimizers of the Riesz-type energy. For X≳N1/2 both upper and lower bound match for maximally-separated point sets satisfying ‖xi−xj‖≳N−1/2.