Spectral Measures Associated with the Factorization of the Lebesgue Measure on a Set via Convolution

Abstract

Abstract. Let Q be a fundamental domain of some full-rank lattice in R d and let µ and ν be two positive Borel measures on R d such that the convolution µ * ν is a multiple of χ Q . We consider the problem as to whether or not both measures must be spectral (i.e. each of their respective associated L 2 space admits an orthogonal basis of exponentials) and we show that this is the case when Q = [0, 1] d . This theorem yields a large class of examples of spectral measures which are either absolutely continuous, singularly continuous or purely discrete spectral measures. In addition, we propose a generalized Fuglede's conjecture for spectral measures on R 1 and we show that it implies the classical Fuglede's conjecture on R 1 .

Perfect fractal sets with zero Fourier dimension and arbitrarily long arithmetic progressions

“…We also remark that special cases of the Moran measures were considered by the author in the study of fractal measures with exponential bases and frames [8,12].…”