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Bad(s,t) is hyperplane absolute winning

Erez Nesharim;  David Simmons — 2014

Acta Arithmetica

J. An proved that for any s,t ≥ 0 such that s + t = 1, Bad (s,t) is (34√2)¯¹-winning for Schmidt's game. We show that using the main lemma from [An] one can derive a stronger result, namely that Bad (s,t) is hyperplane absolute winning in the sense of [BFKRW]. As a consequence, one can deduce the full Hausdorff dimension of Bad (s,t) intersected with certain fractals.

Diophantine approximation in Banach spaces

Lior Fishman;  David Simmons;  Mariusz Urbański — 2014

Journal de Théorie des Nombres de Bordeaux

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 define what it means for such a theorem to be optimal. We show that optimality is implied by but does not imply the existence of badly approximable points.

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