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On generalized Thue-Morse functions and their values (2019)
Journal Article
Badziahin, D., & Zorin, E. (2020). On generalized Thue-Morse functions and their values. Journal of the Australian Mathematical Society, 108(2), 177-201. https://doi.org/10.1017/s1446788718000459

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,... Read More about On generalized Thue-Morse functions and their values.

Finding special factors of values of polynomials at integer points (2016)
Journal Article
Badziahin, D. (2017). Finding special factors of values of polynomials at integer points. International Journal of Number Theory, 13(01), 209-228. https://doi.org/10.1142/s1793042117500129

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... Read More about Finding special factors of values of polynomials at integer points.

An Unusual Continued Fraction (2015)
Journal Article
Badziahin, D., & Shallit, J. (2016). An Unusual Continued Fraction. Proceedings of the American Mathematical Society, 144(5), 1887-1896. https://doi.org/10.1090/proc/12848

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.... Read More about An Unusual Continued Fraction.

On the complexity of a putative counterexample to the p-adic Littlewood conjecture (2015)
Journal Article
Badziahin, D., Bugeaud, Y., Einsiedler, M., & Kleinbock, D. (2015). On the complexity of a putative counterexample to the p-adic Littlewood conjecture. Compositio Mathematica, 151(09), 1647-1662. https://doi.org/10.1112/s0010437x15007393

Let ∥⋅∥ denote the distance to the nearest integer and, for a prime number p, let |⋅|p denote the p-adic absolute value. Over a decade ago, de Mathan and Teulié [Problèmes diophantiens simultanés, Monatsh. Math. 143 (2004), 229–245] asked whether inf... Read More about On the complexity of a putative counterexample to the p-adic Littlewood conjecture.

Thue-Morse constant is not badly approximable (2014)
Journal Article
Badziahin, D., & Zorin, E. (2015). Thue-Morse constant is not badly approximable. International Mathematics Research Notices, 2015(19), 9618-9637. https://doi.org/10.1093/imrn/rnu238

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... Read More about Thue-Morse constant is not badly approximable.

Badly approximable points on planar curves and a problem of Davenport (2014)
Journal Article
Badziahin, D., & Velani, S. (2014). Badly approximable points on planar curves and a problem of Davenport. Mathematische Annalen, 359(3-4), 969-1023. https://doi.org/10.1007/s00208-014-1020-z

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... Read More about Badly approximable points on planar curves and a problem of Davenport.

Inhomogeneous theory of dual Diophantine approximation on manifolds (2012)
Journal Article
Badziahin, D., Beresnevich, V., & Velani, S. (2013). Inhomogeneous theory of dual Diophantine approximation on manifolds. Advances in Mathematics, 232(1), 1-35. https://doi.org/10.1016/j.aim.2012.09.022

The theory of inhomogeneous Diophantine approximation on manifolds is developed. In particular, the notion of nice manifolds is introduced and the divergence part of the Groshev type theory is established for all such manifolds. Our results naturally... Read More about Inhomogeneous theory of dual Diophantine approximation on manifolds.

Multiplicatively badly approximable numbers and generalised Cantor sets (2011)
Journal Article
Badziahin, D., & Velani, S. (2011). Multiplicatively badly approximable numbers and generalised Cantor sets. Advances in Mathematics, 228(5), 2766-2796. https://doi.org/10.1016/j.aim.2011.06.041

Let p be a prime number. The p -adic case of the Mixed Littlewood Conjecture states that View the MathML sourceliminfq→∞q⋅|q|p⋅‖qα‖=0 for all α∈Rα∈R. We show that with the additional factor of View the MathML sourcelogqloglogq the statement is false.... Read More about Multiplicatively badly approximable numbers and generalised Cantor sets.

On a problem in simultaneous Diophantine approximation: Schmidt's conjecture (2011)
Journal Article
Badziahin, D., Pollington, A., & Velani, S. (2011). On a problem in simultaneous Diophantine approximation: Schmidt's conjecture. Annals of Mathematics, 174(3), 1837-1883. https://doi.org/10.4007/annals.2011.174.3.9

For any i,j≥0 with i+j=1 , let Bad(i,j) denote the set of points (x,y)∈R 2 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 has full dimension... Read More about On a problem in simultaneous Diophantine approximation: Schmidt's conjecture.

The mixed Schmidt conjecture in the theory of Diophantine approximation (2011)
Journal Article
Badziahin, D., Levesley, J., & Velani, S. (2011). The mixed Schmidt conjecture in the theory of Diophantine approximation. Mathematika, 57(02), 239-245. https://doi.org/10.1112/s0025579311002075

Let xs1D49F=(dn)∞n=1 be a sequence of integers with dn≥2, and let (i,j) be a pair of strictly positive numbers with i+j=1. We prove that the set of xxs2208xs211D for which there exists some constant c(x)≧0 such that \[ \max \!\big \{|q|_\mathcal {D}^... Read More about The mixed Schmidt conjecture in the theory of Diophantine approximation.

Inhomogeneous Diophantine approximation on curves and Hausdorff dimension (2010)
Journal Article
Badziahin, D. (2010). Inhomogeneous Diophantine approximation on curves and Hausdorff dimension. Advances in Mathematics, 223(1), 329-351. https://doi.org/10.1016/j.aim.2009.08.005

The goal of this paper is to develop a coherent theory for inhomogeneous Diophantine approximation on curves in RnRn akin to the well established homogeneous theory. More specifically, the measure theoretic results obtained generalize the fundamental... Read More about Inhomogeneous Diophantine approximation on curves and Hausdorff dimension.