In this paper we study the values of modular functions at the Markov quadratics which are defined in terms of their cycle integrals along the associated closed geodesics. These numbers are shown to satisfy two properties that were conjectured by Kaneko. More precisely we show that the values of a modular function f , along any branch B of the Markov tree, converge to the value of f at the Markov number which is the predecessor of the tip of B. We also prove an interlacing property for these values.

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This paper is motivated by Davenport's problem and the subsequent work regarding badly approximable points in submanifolds of a Euclidian space. We study the problem in the area of twisted Diophantine approximation and present two different approaches. The first approach shows that, under a certain restriction, any countable intersection of the sets of weighted badly approximable points on any non-degenerate C 1 submanifold of R n has full dimension. In the second approach we introduce the property of isotropically winning and show that the sets of weighted badly approximable points are isotropically winning under the same restriction as above.

We will give new upper bounds for the number of solutions to the inequalities of the shape |F (x, y)| ≤ h, where F (x, y) is a sparse binary form, with integer coefficients, and h is a sufficiently small integer in terms of the discriminant of the binary form F . Our bounds depend on the number of non-vanishing coefficients of F (x, y). When F is "really sparse", we establish a sharp upper bound for the number of solutions that is linear in terms of the number of non-vanishing coefficients. This work will provide affirmative answers to a number of conjectures posed by Mueller and Schmidt in [18] in special but important cases.2000 Mathematics Subject Classification. 11D45.

For any j 1 , . . . , j n > 0 with n i=1 j i = 1 and any θ ∈ R n , let Bad θ (j 1 , . . . , j n ) denote the set of points η ∈ R n for which max 1≤i≤n ( qθ i − η i 1/ji ) > c/q for some positive constant c = c(η) and all q ∈ N. These sets are the 'twisted' inhomogeneous analogue of Bad(j 1 , . . . , j n ) in the theory of simultaneous Diophantine approximation. It has been shown that they have full Hausdorff dimension in the non-weighted setting, i.e provided that j i = 1/n, and in the weighted setting when θ is chosen from Bad(j 1 , . . . , j n ). We generalise these results proving the full Hausdorff dimension in the weighted setting without any condition on θ. Moreover, we prove dim(Bad θ (j 1 , . . . , j n ) ∩ Bad(1, 0, . . . , 0) ∩ . . . ∩ Bad(0, . . . , 0, 1)) = n.