School of Mathematics and Statistics |
Dmitry BadziahinAssociate Professor at the Univesity of SydneyContact details
Office C634
School of Mathematics and Statistics F07 University of Sydney NSW 2006 Australia Phone: +61 2 9351 3279 Email: dzmitry.badziahin@sydney.edu.au |
My research interests lie mostly in Analytical Number theory, in particular Diophantine approximation. More specific areas of interest:
D. Badziahin, A. Gorodnik and N. Peyerimhoff, Dynamics and Analytic Number theory, London Mathematical Society Lecture Note Series 437, Cambridge University Press, 2016. Written by leading experts, this book explores several directions of current research at the interface between dynamics and analytic number theory. Topics include Diophantine approximation, exponential sums, Ramsey theory, ergodic theory and homogeneous dynamics. The origins of this material lie in the 'Dynamics and Analytic Number Theory' Easter School held at Durham University in 2014. Key concepts, cutting-edge results, and modern techniques that play an essential role in contemporary research are presented in a manner accessible to young researchers, including PhD students. This book will also be useful for established mathematicians. The areas discussed include ubiquitous systems and Cantor-type sets in Diophantine approximation, flows on nilmanifolds and their connections with exponential sums, multiple recurrence and Ramsey theory, counting and equidistribution problems in homogeneous dynamics, and applications of thin groups in number theory. Both dynamical and 'classical' approaches towards number theoretical problems are also provided. |
Simultaneous Diophantine approximation to points on the Veronese curve.
With C. Eggins, Badly approximable infinite products of quadratic polynomials.
With S. Harrap, E. Nesharim, D. Simmons, Schmidt games and Cantor winning sets.
On continued fraction expansion of potential counterexamples to p-adic Littlewood conjecture.
On effective irrationality exponents of cubic irrationals, Ramanujan J., 2024.
Continued fractions of cubic Laurent series, Annali della Scuola Normale Superiore di Pisa, 2023.
With J. Shallit, Badly approximable numbers, Kronecker's theorem, and diversity of Sturmian characteristic sequences, Journal de Theorie des Nombres de Bordeaux, V. 35 (1), pp. 1-15, 2023.
Upper bounds for the uniform simultaneous Diophantine exponents, Mathematika, V.68 (3), pp. 805 -- 826, 2022.
With Y. Bugeaud, Multiplicative p-adic approximation, Michigan Math. J., V. 71(1), pp. 121 -- 143, 2022.
With Y. Bugeaud, J. Schleischitz, On simultaneous rational approximation to a p-adic number and its integral powers, II, Proc. Edinburgh Math. Soc., V. 64(2), pp. 317 -- 337, 2021.
On $t$-adic Littlewood conjecture for generalised Thue-Morse functions, Proc. AMS, V. 149(11), pp. 4527 -- 4540, 2021.
With J. Schleischitz, An improved bound in Wirsing's problem, Tran. AMS, V. 374(3), pp. 1847 -- 1861, 2021.
With Y. Bugeaud, On simultaneous rational approximation to a real number and its integral powers, II, New York J. Math., V. 26, pp. 362 -- 377, 2020.
With E. Zorin, On generalized Thue-Morse functions and their values, J. of AustMS., V. 108(2), pp. 177 -- 201, 2020.
With E. Zorin, On irrationality measure of Thue-Morse constant, Math. Proc. Cambridge Philos. Soc., V. 168(3), pp. 455 -- 472, 2020.
On spectrum of irrationality exponents of Mahler numbers. J. de Theorie des Nombres de Bordeaux, V 31(2), pp. 431 -- 453, 2019.
Continued fractions of certain Mahler functions, Acta Arithmetica, V. 188(1), pp. 53 -- 81, 2019.
With S. Harrap, Cantor-winning sets and their applications, Advances in Math., V. 318, pp. 627 -- 677, 2017.
With S. Harrap, M. Hussain, An Inhomogeneous Jarnik type theorem for planar curves, Math. Proc. Cambridge. Philos. Soc., V 163(1), pp. 47 -- 70, 2017.
Finding special factors of values of polynomials at integer points, Int. J. of Numb. Theor., V. 13(1), 2017.
Continuant Diophantine equations, Moscow J. of Comb. and Numb. Theor., V 6(2--3), pp. 38 -- 55, 2016.
Computation of the Infimum in the Littlewood Conjecture, Experimental Math., V. 25(1), pp. 100 -- 105, 2016.
With J. Shallit, An Unusual Continued Fraction, Proc. Amer. Math. Soc., V. 144(5), pp. 1887 -- 1896, 2015.
With Y. Bugeaud, M. Einsiedler, D. Kleinbock, On the complexity of a putative counterexample to the p-adic Littlewood conjecture, Compositio Math., V. 151(9), pp. 1647 -- 1662, 2015.
With E. Zorin, Thue-Morse constant is not badly approximable, Int. Math. Res. Notes., V. 2015(19), pp. 9618 -- 9637, 2015.
With S. Velani, Badly approximable points on planar curves and a problem of Davenport , Math. Annalen, V. 359(3--4), pp. 969 -- 1023, 2014.
On multiplicatively badly approximable numbers, Mathematika, V. 59(1), pp. 31 -- 55, 2013.
With V. Beresnevich and S. Velani, Inhomogeneous theory of dual Diophantine approximation on manifolds, Advances in Math., V. 232, pp. 1 -- 35, 2013.
With S. Velani. Multiplicatively badly approximable numbers and generalised Cantor sets, Advances in Math., V. 228(5), pp. 2766 -- 2796, 2011.
With A. Pollington and S. Velani, On a problem in simultaneous Diophantine approximation: Schmidt's conjecture, Annals of Math., V. 174, pp. 1837 -- 1883, 2011.
With J. Levesley and S. Velani, The mixed Schmidt conjecture in the theory of Diophantine approximation, Mathematika, V. 57(2), pp. 239 -- 245, 2011.
With A. Haynes, A note on Farey fractions with denominators in arithmetic progressions, Acta Arithmetica, V. 147(3), pp. 205 -- 215, 2011.
Inhomogeneous Diophantine approximation on curves and Hausdorff dimension, Advances in Math., V. 223(1), pp. 329 -- 351, 2009.
With I. Morozova, On the exact exponent of convergence of series with small denominators. Vestsi Nats. Akad. Navuk Belarusi Ser. Fiz.-Mat. Navuk 2007, N. 2, pp 37 -- 40. (in Russian)
With J. Levesley, Convergence results for simultaneous and multiplicative Diophantine approximation on planar curves, Glasgow Math. J., V. 49, pp. 367 -- 375, 2007.
With J. Levesley, Computation of fractional measures in the case of convergence for Diophantine approximations on planar curves. Dokl. Nats. Akad. Nauk Belarusi V. 50, N. 6, pp. 24--27, 2006. (in Russian)
Inhomogeneous Diophantine approximation and lower bounds of Hausdorff dimension, Vesti Nats. Acad. Navuk Belarusi. Ser. Fiz.-Mat. Navuk 2005, N. 3, pp 32 -- 36. (in Russian)
With I. Morozova, On small denominators of product of linear forms, proceeding of the international mathematical conference in honor of Skorobogatko, Lvov 2004. (in Russian)
Here are the computational projects in which I participated: