Thue-Morse constant is not badly approximable

We prove that Thue–Morse constant τTM=0.01101001…2 is not a badly approximable number. Moreover, we prove that τTM(a)=0.01101001…a is not badly approximable for every integer base a≥2 such that a is not divisible by 15. At the same time, we provide a precise formula for convergents of the Laurent series f~TM(z)=z−1∏∞n=1(1−z−2n), thus developing further the research initiated by Alf van der Poorten and others

Asymptotic diophantine approximation:the multiplicative case

Let $\alpha$ and $\beta$ be irrational real numbers and 0<\F<1/30. We prove a precise estimate for the number of positive integers $q\leq Q$ that satisfy \|q\alpha\|\cdot\|q\beta\|<\F. If we choose \F as a function of $Q$ we get asymptotics as $Q$ gets large, provided \F Q grows quickly enough in terms of the (multiplicative) Diophantine type of $(\alpha,\beta)$, e.g., if $(\alpha,\beta)$ is a counterexample to Littlewood's conjecture then we only need that \F Q tends to infinity. Our result yields a new upper bound on sums of reciprocals of products of fractional parts, and sheds some light on a recent question of L\^{e} and Vaaler.Comment: To appear in Ramanujan Journa

Badly approximable points on manifolds

Addressing a problem of Davenport we show that any finite intersection of the sets of weighted badly approximable points on any analytic nondegenerate manifold in has full dimension. This also extends Schmidt's conjecture on badly approximable points to arbitrary dimensions

Finding special factors of values of polynomials at integer points

We investigate the divisors dd of the numbers P(n)P(n) for various polynomials P∈Z[x]P∈ℤ[x] such that d≡1(modn)d≡1(modn). We obtain the complete classification of such divisors for a class of polynomials, in particular for P(x)=x4+1P(x)=x4+1. We also construct a fast algorithm which provides all such factorizations up to a given limit for another class, for example for P(x)=2x4+1P(x)=2x4+1. We use these results to find all the divisors d=2mk+1d=2mk+1 of numbers 24m+124m+1 and 24m+1+124m+1+1. For the numbers 24m+124m+1 the complete classification of such divisors is provided while for the numbers 24m+1+124m+1+1 the given classification is proved to be exhaustive only for m≤1000m≤1000

Cantor-winning sets and their applications

We introduce and develop a class of Cantor-winning sets that share the same amenable properties as the classical winning sets associated to Schmidt’s (α, β)-game: these include maximal Hausdorff dimension, invariance under countable intersections with other Cantor-winning sets and invariance under bi-Lipschitz homeomorphisms. It is then demonstrated that a wide variety of badly approximable sets appearing naturally in the theory of Diophantine approximation fit nicely into our framework. As applications of this phenomenon we answer several previously open questions, including some related to the Mixed Littlewood conjecture and the ×2,×3 problem

An Unusual Continued Fraction

We consider the real number σ with continued fraction expansion [a0, a1, a2,...] = [1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 16,...], where ai is the largest power of 2 dividing i + 1. We show that the irrationality measure of σ2 is at least 8/3. We also show that certain partial quotients of σ2 grow doubly exponentially, thus confirming a conjecture of Hanna and Wilson

Badly approximable points on planar curves and a problem of Davenport

Let C be two times continuously differentiable curve in R2 with at least one point at which the curvature is non-zero. For any i,j⩾0 with i+j=1, let Bad(i,j) denote the set of points (x,y)∈R2 for which max{∥qx∥1/i,∥qy∥1/j}>c/q for all q∈N. Here c=c(x,y) is a positive constant. Our main result implies that any finite intersection of such sets with C has full Hausdorff dimension. This provides a solution to a problem of Davenport dating back to the sixties

On generalized Thue-Morse functions and their values

In this paper we extend and generalize, up to a natural bound of the method, our previous work Badziahin and Zorin [‘Thue–Morse constant is not badly approximable’, Int. Math. Res. Not. IMRN 19 (2015), 9618–9637] where we proved, among other things, that the Thue–Morse constant is not badly approximable. Here we consider Laurent series defined with infinite products fd(x) = Q∞ n=0 (1 − x −d n ), d ∈ N, d ≥ 2, which generalize the generating function f2(x) of the Thue–Morse number, and study their continued fraction expansion. In particular, we show that the convergents of x −d+1 fd(x) have a regular structure. We also address the question of whether the corresponding Mahler numbers fd(a) ∈ R, a, d ∈ N, a, d ≥ 2, are badly approximable