Vertical Shift and Simultaneous Diophantine Approximation on Polynomial Curves

TY - JOUR

T1 - Vertical Shift and Simultaneous Diophantine Approximation on Polynomial Curves

AU - Adiceam, Faustin

PY - 2015/2

Y1 - 2015/2

N2 - The Hausdorff dimension of the set of simultaneously τ-well-approximable points lying on a curve defined by a polynomial P(X) + α, where P(X) ∈ ℤ[X] and α ∈ ℝ, is studied when τ 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 that 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.

AB - The Hausdorff dimension of the set of simultaneously τ-well-approximable points lying on a curve defined by a polynomial P(X) + α, where P(X) ∈ ℤ[X] and α ∈ ℝ, is studied when τ 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 that 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.

KW - diophantine approximation

KW - manifolds

KW - metric theory

KW - polynomial curves

UR - http://www.scopus.com/inward/record.url?scp=84937938426&partnerID=8YFLogxK

U2 - 10.1017/S0013091514000169

DO - 10.1017/S0013091514000169

M3 - Article

AN - SCOPUS:84937938426

SN - 0013-0915

VL - 58

JO - Proceedings of the Edinburgh Mathematical Society

JF - Proceedings of the Edinburgh Mathematical Society

IS - 1

ER -