arXiv subjects

Lior Fishman

Publications and source records attributed to Lior Fishman.

41 records · Page 3Linked to original sources

arXiv2011

Intrinsic Approximation on Cantor-like Sets, a Problem of Mahler

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 such a theory, we prove a Dirichlet-type theorem for this intrinsic diophantine approximation on Cantor-like sets, and discuss related possible theorems/conjectures. The resulting approximation function is analogous to that for R^d, but with d being the Hausdorff dimension of the set, and logarithmic dependence on the denominator instead.

math.NT↗

arXiv2011

The set of badly approximable vectors is strongly $C^1$ incompressible

We prove that the countable intersection of $C^1$-diffeomorphic images of certain Diophantine sets has full Hausdorff dimension. For example, we show this for the set of badly approximable vectors in $\mathbb{R}^d$, improving earlier results of Schmidt and Dani. To prove this, inspired by ideas of McMullen, we define a new variant of Schmidt's $(α, β)$-game and show that our sets are hyperplane absolute winning (HAW), which in particular implies winning in the original game. The HAW property passes automatically to games played on certain fractals, thus our sets intersect a large class of fractals in a set of positive dimension. This extends earlier results of Fishman to a more general set-up, with simpler proofs.

math.NT↗

arXiv2011

Diophantine approximations on fractals

We exploit dynamical properties of diagonal actions to derive results in Diophantine approximations. In particular, we prove that the continued fraction expansion of almost any point on the middle third Cantor set (with respect to the natural measure) contains all finite patterns (hence is well approximable). Similarly, we show that for a variety of fractals in [0,1]^2, possessing some symmetry, almost any point is not Dirichlet improvable (hence is well approximable) and has property C (after Cassels). We then settle by similar methods a conjecture of M. Boshernitzan saying that there are no irrational numbers x in the unit interval such that the continued fraction expansions of {nx mod1 : n is a natural number} are uniformly eventually bounded.

math.DS↗

arXiv2010

Schmidt's Game on Certain Fractals

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 the Hausdorff dimension for some. In particular we prove that if K is the attractor of an irreducible finite family of contracting similarity maps of R^N satisfying the open set condition then for any countable collection of non-singular affine transformations Λ_i:R^N \to R^N, dimK=dimK\cap (\cap ^{\infty}_{i=1}(Λ_i(BA))) where BA is the set of badly approximable vectors in R^N.

math.NT↗

arXiv2008

Schmidt's game, Badly Approximable Linear Forms and Fractals

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 winning set in Schmidt's game sense played on K, where BA(M,N) is the set of badly approximable M\times N matrices. As an immediate consequence we have the following application. If K is the attractor of an irreducible finite family of contracting similarity maps of R^(M\times N) satisfying the open set condition, (the Cantor ternary set, Koch's curve and Sierpinski's gasket to name a few examples), the dimK=dimK\capBA(M,N).

math.NT↗