9 research outputs found
Bad(s,t) is hyperplane absolute winning
Get PDFJ. An (2013) proved that for any such that , is -winning for Schmidt's game. We show that using the main lemma from An's paper one can derive a stronger result, namely that is hyperplane absolute winning in the sense of Broderick, Fishman, Kleinbock, Reich, and Weiss (2012). As a consequence one can deduce the full dimension of intersected with certain fractals
Bad(w) is hyperplane absolute winning
Get PDFIn 1998 Kleinbock conjectured that any set of weighted badly approximable real matrices is a winning subset in the sense of Schmidt's game. In this paper we prove this conjecture in full for vectors in in arbitrary dimensions by showing that the corresponding set of weighted badly approximable vectors is hyperplane absolute winning. The proof uses the Cantor potential game played on the support of Ahlfors regular absolutely decaying measures and the quantitative nondivergence estimate for a class of fractal measures due to Kleinbock, Lindenstrauss and Weiss. To establish the existence of a relevant winning strategy in the Cantor potential game we introduce a new approach using two independent diagonal actions on the space of lattices
On the t-adic Littlewood Conjecture
No full textThe p–adic Littlewood Conjecture due to De Mathan and Teulié asserts that for any prime number p and any real number α, the equation infm|≥1|m| · |m|p · |hmαi| = 0 holds. Here, |m| is the usual absolute value of the integer m, |m|p its p–adic absolute value and |hxi| denotes the distance from a real number x to the set of integers. This still open conjecture stands as a variant of the well–known Littlewood Conjecture. In the same way asthe latter, it admits a natural counterpart over the field of formal Laurent series Kt−1of a ground field K. This is the so–called tadic Littlewood Conjecture (t–LC).It is known that t–LC fails when the ground field K is infinite. This article is concernedwith the much more difficult case when the latter field is finite. More precisely, a fully explicitcounterexample is provided to show that t–LC does not hold in the case that K is a finite field with characteristic 3. Generalizations to fields with characteristics different from 3 arealso discussed.The proof is computer assisted. It reduces to showing that an infinite matrix encoding Hankel determinants of the Paper–Folding sequence over F3, the so–called Number Wall of thissequence, can be obtained as a two–dimensional automatic tiling satisfying a finite number of suitable local constrai
On the t-adic Littlewood Conjecture
No full textThe p–adic Littlewood Conjecture due to De Mathan and Teulié asserts that for any prime number p and any real number α, the equation infm|≥1|m| · |m|p · |hmαi| = 0 holds. Here, |m| is the usual absolute value of the integer m, |m|p its p–adic absolute value and |hxi| denotes the distance from a real number x to the set of integers. This still open conjecture stands as a variant of the well–known Littlewood Conjecture. In the same way asthe latter, it admits a natural counterpart over the field of formal Laurent series Kt−1of a ground field K. This is the so–called tadic Littlewood Conjecture (t–LC).It is known that t–LC fails when the ground field K is infinite. This article is concernedwith the much more difficult case when the latter field is finite. More precisely, a fully explicitcounterexample is provided to show that t–LC does not hold in the case that K is a finite field with characteristic 3. Generalizations to fields with characteristics different from 3 arealso discussed.The proof is computer assisted. It reduces to showing that an infinite matrix encoding Hankel determinants of the Paper–Folding sequence over F3, the so–called Number Wall of thissequence, can be obtained as a two–dimensional automatic tiling satisfying a finite number of suitable local constrai
Schmidt's game on Hausdorff metric and function spaces : generic dimension of sets and images
Get PDFAF was financially supported by an ERC Consolidator Grant (772466) and by The MTA Momentum Project (LP2016‐5). JMF was financially supported by a Leverhulme Trust Research Fellowship (RF‐2016‐500) and an EPSRC Standard Grant (EP/R015104/1). EN and DS were supported by an EPSRC Programme Grant (EP/J018260/1).We consider Schmidt's game on the space of compact subsets of a given metric space equipped with the Hausdorff metric, and the space of continuous functions equipped with the supremum norm. We are interested in determining the generic behavior of objects in a metric space, mostly in the context of fractal dimensions, and the notion of “generic” we adopt is that of being winning for Schmidt's game. We find properties whose corresponding sets are winning for Schmidt's game that are starkly different from previously established, and well‐known, properties which are generic in other contexts, such as being residual or of full measure.Peer reviewe
Low Discrepancy Digital Kronecker-Van der Corput Sequences
No full textThe discrepancy of a sequence measures how quickly it approaches a uniform distribution. Given a natural number d, any collection of one-dimensional so-called low discrepancy sequences {Si : 1 ≤ i ≤ d} can be concatenated to create a d-dimensional hybrid sequence (S1, . . . , Sd). Since their introduction by Spanier in 1995, many connections between the discrepancy of a hybrid sequence and the discrepancy of its component sequences have been discovered. However, a proof that a hybrid sequence is capable of being low discrepancy has remained elusive. The purpose of this note is to remedy this by providing an explicit connection between Diophantine approximation over function fields and two dimensional low discrepancy hybrid sequences.Specifically, let Fq be the finite field of cardinality q. It is shown that some real numbered hybrid sequence H(Θ(t), P(t)) := H(Θ, P) built from the digital Kronecker sequence associated to a Laurent series Θ(t) ∈ Fq((t−1)) and the digital Van der Corput sequence associated to an irreducible polynomial P(t) ∈ Fq[t] meets the above property. More precisely, if Θ(t) is a counterexample to the so called t-adic Littlewood Conjecture (t-LC), then another Laurent series Φ(t) ∈ Fq((t−1)) induced from Θ(t) and P(t) can be constructed so that H(Φ, P) is low discrepancy. Such counterexamples to t-LC are known over a number of finite fields by, on the one hand, Adiceam, Nesharim and Lunnon, and on the other, by Garrett and the author.<div/
Low Discrepancy Digital Kronecker-Van der Corput Sequences
Get PDFThe discrepancy of a sequence measures how quickly it approaches a uniform distribution. Given a natural number , any collection of one-dimensional so-called low discrepancy sequences can be concatenated to create a -dimensional . Since their introduction by Spanier in 1995, many connections between the discrepancy of a hybrid sequence and the discrepancy of its component sequences have been discovered. However, a proof that a hybrid sequence is capable of being low discrepancy has remained elusive. This paper remedies this by providing an explicit connection between Diophantine approximation over function fields and two dimensional low discrepancy hybrid sequences.
Specifically, let be the finite field of cardinality . It is shown that some real numbered hybrid sequence built from the digital Kronecker sequence associated to a Laurent series and the digital Van der Corput sequence associated to an irreducible polynomial meets the above property. More precisely, if is a counterexample to the so called (-), then another Laurent series induced from and can be constructed so that is low discrepancy. Such counterexamples to - are known over a number of finite fields by, on the one hand, Adiceam, Nesharim and Lunnon, and on the other, by Garrett and the author.13 pages, 1 figur
