Diophantine approximation on planar curves: the convergence theory

A note on simultaneous Diophantine approximation on planar curves

Let Sn(ψ1, . . . , ψn) denote the set of simultaneously (ψ1, . . . , ψn)–approximable points in Rn and S∗ n (ψ) denote the set of multiplicatively ψ–approximable points in Rn. Let M be a manifold in Rn. The aim is to develop a metric theory for the sets M∩Sn(ψ1, . . . , ψn) and M∩S∗ n (ψ) analogous to the classical theory in whichM is simply Rn. In this note, we mainly restrict our attention to...

Simultaneous Diophantine Approximation on Planar

Let C be a non-degenerate planar curve. We show that the curve is of Khintchine-type for convergence in the case of simultaneous approximation in R 2 with two independent approximation functions; that is if a certain sum converges then the set of all points (x, y) on the curve which satisfy simultaneously the inequalities qx < ψ1(q) and qy < ψ2(q) infinitely often has induced measure 0. This co...

Diophantine Approximation and Algebraic Curves

The first topic of the workshop, Diophantine approximation, has at its core the study of rational numbers which closely approximate a given real number. This topic has an ancient history, going back at least to the first rational approximations for π. The adjective Diophantine comes from the third century Hellenistic mathematician Diophantus, who wrote an influential text solving various equati...

Diophantine approximation on planar curves and the distribution of rational points

Let C be a non–degenerate planar curve and for a real, positive decreasing function ψ let C(ψ) denote the set of simultaneously ψ–approximable points lying on C. We show that C is of Khintchine type for divergence; i.e. if a certain sum diverges then the one-dimensional Lebesgue measure on C of C(ψ) is full. We also obtain the Hausdorff measure analogue of the divergent Khintchine type result. ...

Diophantine Approximation on Manifolds and the Distribution of Rational Points: Contributions to the Convergence Theory