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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.