Badly approximable points on planar curves and a problem of Davenport

TY - JOUR

T1 - Badly approximable points on planar curves and a problem of Davenport

AU - Badziahin, Dzmitry

AU - Velani, Sanju

N1 - 44 pages

PY - 2014/8

Y1 - 2014/8

N2 - 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.

AB - 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.

KW - math.NT

KW - 11J83, 11J13, 11K60

U2 - 10.1007/s00208-014-1020-z

DO - 10.1007/s00208-014-1020-z

M3 - Article

SN - 1432-1807

VL - 359

SP - 969

EP - 1023

JO - Mathematische Annalen

JF - Mathematische Annalen

IS - 3-4

ER -