2019/04/09 by Ferguson, Sam, Mayeli, Azita, Sothanaphan, Nat
#11L07 (Secondary) #43A40 (Primary) #43A70 #52C22 #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1904.04487
Motivated by the open problem of exhibiting a subset of Euclidean space which has no exponential Riesz basis, we focus on exponential Riesz bases in finite abelian groups. We point out that that every subset of a finite abelian group has such a basis, removing interest in the existence question in this context. We then define tightness quantities for subsets to measure the conditioning of Riesz bases; for normalized tightness quantities, a value of one corresponds to an orthogonal basis, and a value of infinity corresponds to nonexistence of a basis. As an application, we obtain new weak evidence in favor of the open problem by giving a sequence of subsets of finite abelian groups whose tightness quantities go to infinity in the limit. We also prove that the Cartesian product of a set with a finite abelian group has the same tightness quantities as the original set. Lastly, under an additional hypothesis, explicit bounds are given for tightness quantities in terms of a subset's lowest multi-tiling level by a subgroup and its geometric configuration. This establishes a quantitative link between discrete geometry and harmonic analysis in this setting.