- A fractal uncertainty principle for Bergman spaces and analytic wavelets(arXiv)
Abstract : Motivated by results of Dyatlov on Fourier uncertainty principles for Cantor sets and of Knutsen, for joint time-frequency representations (STFT with Gaussian, equivalent to Fock spaces), we suggest a general setting relating localization and uncertainty and prove, within this context, an uncertainty principle for Cantor sets in Bergman spaces of the unit disc, defined as unions of annuli equidistributed in the hyperbolic measure. The result can be written in terms of analytic Cauchy wavelets. As in the case of the STFT considered by Knutsen, our result consists of a double side bound for the norm of a localization operator involving the fractal dimension log2/log3 in the exponent. As in the STFT case and in Dyatlov’s fractal uncertainty principle, the norm of the dilated iterates of the Cantor set in the disc tends to infinite, while the corresponding norm of the localization operator tends to zero.
2. Additive energy of regular measures in one and higher dimensions, and the fractal uncertaintyprinciple(arXiv)
Abstract : We obtain new bounds on the additive energy of (Ahlfors-David type) regular measures in both one and higher dimensions, which implies expansion results for sums and products of the associated regular sets, as well as more general nonlinear functions of these sets. As a corollary of the higher-dimensional results we obtain some new cases of the fractal uncertainty principle in odd dimensions.
3. An introduction to fractal uncertainty principle(arXiv)
Author : Semyon Dyatlov
Abstract : Fractal uncertainty principle states that no function can be localized in both position and frequency near a fractal set. This article provides a review of recent developments on the fractal uncertainty principle and of their applications to quantum chaos, including lower bounds on mass of eigenfunctions on negatively curved surfaces and spectral gaps on convex co-compact hyperbolic surfaces.
4.Spectral gaps, additive energy, and a fractal uncertainty principle (arXiv)
Abstract : We obtain an essential spectral gap for n-dimensional convex co-compact hyperbolic manifolds with the dimension δof the limit set close to (n−1)/2. The size of the gap is expressed using the additive energy of stereographic projections of the limit set. This additive energy can in turn be estimated in terms of the constants in Ahlfors-David regularity of the limit set. Our proofs use new microlocal methods, in particular a notion of a fractal uncertainty principle