Badly approximable points on planar curves and a problem of Davenport

TL;DR

This paper proves that the intersection of certain Diophantine approximation sets with a smooth planar curve has full Hausdorff dimension, solving a longstanding problem posed by Davenport in the 1960s.

Contribution

It establishes that finite intersections of Bad(i,j) sets with a non-degenerate planar curve have full Hausdorff dimension, advancing understanding in metric Diophantine approximation.

Findings

Finite intersections with curves have full Hausdorff dimension.

Provides a solution to Davenport's problem from the 1960s.

Extends classical Diophantine approximation results to curved sets.

Abstract

Let C be two times continuously differentiable curve in R^2 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) in R^2 for which max {||qx ||^{1/i}, ||qy||^{1/j}} > c/q for all integers q >0. Here c = c(x,y) is a positive constant. Our main result implies that any finite intersection of such sets with C has full Hausdorff dimension. This provides a solution to a problem of Davenport dating back to the sixties.