83 citations · 293 across the 29 of their papers we have counts for
20 papers · 1 filter
Diophantine approximation on abelian varieties; a conjecture of M. Waldschmidt
Lior Fishman, David Lambert, Keith Merrill +1
Following the work of Waldschmidt, we investigate problems in Diophantine approximation on abelian varieties. First we show that a conjecture of Waldschmidt for a given simple abel…
A variational principle in the parametric geometry of numbers
Tushar Das, Lior Fishman, David Simmons +1
We extend the parametric geometry of numbers (initiated by Schmidt and Summerer, and deepened by Roy) to Diophantine approximation for systems of linear forms in variables,…
A variational principle in the parametric geometry of numbers, with applications to metric Diophantine approximation
Tushar Das, Lior Fishman, David Simmons +1
We establish a new connection between metric Diophantine approximation and the parametric geometry of numbers by proving a variational principle facilitating the computation of the…
Badly approximable vectors and fractals defined by conformal dynamical systems
Tushar Das, Lior Fishman, David Simmons +1
We prove that if is the limit set of an irreducible conformal iterated function system (with either finite or countably infinite alphabet), then the badly approximable vectors…
Intrinsic Diophantine approximation on manifolds: General theory
Lior Fishman, Dmitry Kleinbock, Keith Merrill +1
We investigate the question of how well points on a nondegenerate -dimensional submanifold can be approximated by rationals also lying on , establis…
Decaying and non-decaying badly approximable numbers
Ryan Broderick, Lior Fishman, David Simmons
We call a badly approximable number if, roughly, the Lagrange constants of integer multiples of that number decay as fast as possible. In this terminology, a question of…
Variations on Dirichlet's theorem
Lior Fishman, David Simmons
We give a necessary and sufficient condition for the following property of an integer and a pair : There exist and suc…
The Banach-Mazur-Schmidt game and the Banach-Mazur-McMullen game
Lior Fishman, Vanessa Reams, David Simmons
We introduce two new mathematical games, the Banach-Mazur-Schmidt game and the Banach-Mazur-McMullen game, merging well-known games. We investigate the properties of the games, as…
Hausdorff dimensions of very well intrinsically approximable subsets of quadratic hypersurfaces
Lior Fishman, Keith Merrill, David Simmons
We prove an analogue of a theorem of A. Pollington and S. Velani ('05), furnishing an upper bound on the Hausdorff dimension of certain subsets of the set of very well intrinsicall…
Extrinsic Diophantine approximation on manifolds and fractals
Lior Fishman, David Simmons
Fix , and let be either a real-analytic manifold or the limit set of an iterated function system (for example, could be the Cantor set or…
Intrinsic Diophantine approximation on quadric hypersurfaces
Lior Fishman, Dmitry Kleinbock, Keith Merrill +1
We consider the question of how well points in a quadric hypersurface can be approximated by rational points of . This contrasts with the m…
Unconventional height functions in simultaneous Diophantine approximation
Lior Fishman, David Simmons
Simultaneous Diophantine approximation is concerned with the approximation of a point by points , with a view towards jointly min…
Diophantine approximation in Banach spaces
Lior Fishman, David S. Simmons, Mariusz Urbański
In this paper, we extend the theory of simultaneous Diophantine approximation to infinite dimensions. Moreover, we discuss Dirichlet-type theorems in a very general framework and d…
Badly Approximable Systems of Affine Forms and Incompressibility on Fractals
Ryan Broderick, Lior Fishman, David Simmons
We explore and refine techniques for estimating the Hausdorff dimension of exceptional sets and their diffeomorphic images. Specifically, we use a variant of Schmidt's game to dedu…
Intrinsic approximation for fractals defined by rational iterated function systems - Mahler's research suggestion
Lior Fishman, David Simmons
Following K. Mahler's suggestion for further research on intrinsic approximation on the Cantor ternary set, we obtain a Dirichlet type theorem for the limit sets of rational iterat…
Diophantine properties of measures invariant with respect to the Gauss map
Lior Fishman, David Simmons, Mariusz Urbanski
Motivated by the work of D. Y. Kleinbock, E. Lindenstrauss, G. A. Margulis, and B. Weiss, we explore the Diophantine properties of probability measures invariant under the Gauss ma…
The set of badly approximable vectors is strongly incompressible
Ryan Broderick, Lior Fishman, Dmitry Kleinbock +2
We prove that the countable intersection of -diffeomorphic images of certain Diophantine sets has full Hausdorff dimension. For example, we show this for the set of badly appr…
Intrinsic Approximation on Cantor-like Sets, a Problem of Mahler
Ryan Broderick, Lior Fishman, Asaf Reich
In 1984, Kurt Mahler posed the following fundamental question: How well can irrationals in the Cantor set be approximated by rationals in the Cantor set? Towards development of suc…
Schmidt's game, Badly Approximable Linear Forms and Fractals
Lior Fishman
We prove that for every two natural numbers M and N, if Tau is a Borel, finite, absolutely friendly measure on a compact set K of R^MN, then the intersection of K and BA(M,N) is a…
Schmidt's Game on Certain Fractals
Lior Fishman
We construct (α,β) and α-winning sets in the sense of Schmidt's game, played on the support of certain measures (very friendly and awfully friendly measures) and show how to derive…