Bobkov, S. G.; Ledoux, M.(
, Journal of fourier analysis applications)
null
(Ed.)
We explore upper bounds on Kantorovich transport distances between probability measures on the Euclidean spaces in terms of their Fourier-Stieltjes transforms, with focus on non-Euclidean metrics. The results are illustrated on empirical measures in the optimal matching problem on the real line.
Kleinbock, Dmitry; Rao, Anurag(
, International Mathematics Research Notices)
Abstract We study a norm-sensitive Diophantine approximation problem arising from the work of Davenport and Schmidt on the improvement of Dirichlet’s theorem. Its supremum norm case was recently considered by the 1st-named author and Wadleigh [ 17], and here we extend the set-up by replacing the supremum norm with an arbitrary norm. This gives rise to a class of shrinking target problems for one-parameter diagonal flows on the space of lattices, with the targets being neighborhoods of the critical locus of the suitably scaled norm ball. We use methods from geometry of numbers to generalize a result due to Andersen and Duke [ 1] on measure zero and uncountability of the set of numbers (in some cases, matrices) for which Minkowski approximation theorem can be improved. The choice of the Euclidean norm on $\mathbb{R}^2$ corresponds to studying geodesics on a hyperbolic surface, which visit a decreasing family of balls. An application of the dynamical Borel–Cantelli lemma of Maucourant [ 25] produces, given an approximation function $\psi $, a zero-one law for the set of $\alpha \in \mathbb{R}$ such that for all large enough $t$ the inequality $\left (\frac{\alpha q -p}{\psi (t)}\right )^2 + \left (\frac{q}{t}\right )^2 < \frac{2}{\sqrt{3}}$ has non-trivial integer solutions.
Kollár, Alicia J., Fitzpatrick, Mattias, Sarnak, Peter, and Houck, Andrew A. Line-Graph Lattices: Euclidean and Non-Euclidean Flat Bands, and Implementations in Circuit Quantum Electrodynamics. Retrieved from https://par.nsf.gov/biblio/10177404. Communications in Mathematical Physics . Web. doi:10.1007/s00220-019-03645-8.
Kollár, Alicia J., Fitzpatrick, Mattias, Sarnak, Peter, & Houck, Andrew A. Line-Graph Lattices: Euclidean and Non-Euclidean Flat Bands, and Implementations in Circuit Quantum Electrodynamics. Communications in Mathematical Physics, (). Retrieved from https://par.nsf.gov/biblio/10177404. https://doi.org/10.1007/s00220-019-03645-8
Kollár, Alicia J., Fitzpatrick, Mattias, Sarnak, Peter, and Houck, Andrew A.
"Line-Graph Lattices: Euclidean and Non-Euclidean Flat Bands, and Implementations in Circuit Quantum Electrodynamics". Communications in Mathematical Physics (). Country unknown/Code not available. https://doi.org/10.1007/s00220-019-03645-8.https://par.nsf.gov/biblio/10177404.
@article{osti_10177404,
place = {Country unknown/Code not available},
title = {Line-Graph Lattices: Euclidean and Non-Euclidean Flat Bands, and Implementations in Circuit Quantum Electrodynamics},
url = {https://par.nsf.gov/biblio/10177404},
DOI = {10.1007/s00220-019-03645-8},
abstractNote = {},
journal = {Communications in Mathematical Physics},
author = {Kollár, Alicia J. and Fitzpatrick, Mattias and Sarnak, Peter and Houck, Andrew A.},
}
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