Vertical shift and simultaneous Diophantine approximation on polynomial curves

TL;DR

This paper investigates the Hausdorff dimension of points on polynomial curves with a vertical shift, extending Diophantine approximation results to non-integer coefficient curves and considering congruential constraints.

Contribution

It provides the first results on Hausdorff dimension for well approximable points on polynomial curves with a real shift, beyond integer coefficient polynomials.

Findings

Hausdorff dimension computed for shifted polynomial curves

Extension of Diophantine approximation to congruential constraints

First results for non-integer coefficient polynomial curves

Abstract

The Hausdorff dimension of the set of simultaneously tau well approximable points lying on a curve defined by a polynomial P(X)+alpha, where P(X) is a polynomial with integer coefficients and alpha is in R, is studied when tau is larger than the degree of P(X). This provides the first results related to the computation of the Hausdorff dimension of the set of well approximable points lying on a curve which is not defined by a polynomial with integer coefficients. The proofs of the results also include the study of problems in Diophantine approximation in the case where the numerators and the denominators of the rational approximations are related by some congruential constraint.