TY - JOUR

T1 - Metrical theorems on systems of affine forms

AU - Hussain, Mumtaz

AU - Kristensen, Simon

AU - Simmons, David

PY - 2020/8

Y1 - 2020/8

N2 - In this paper we discuss metric theory associated with the affine (inhomogeneous) linear forms in the so called doubly metric settings within the classical and the mixed setups. We consider the system of affine forms given by q↦qX+α, where q∈Zm (viewed as a row vector), X is an m×n real matrix and α∈Rn. The classical setting refers to the dist(qX+α,Zm) to measure the closeness of the integer values of the system (X,α) to integers. The absolute value setting is obtained by replacing dist(qX+α,Zm) with dist(qX+α,0); and the more general mixed settings are obtained by replacing dist(qX+α,Zm) with dist(qX+α,Λ), where Λ is a subgroup of Zm. We prove the Khintchine–Groshev and Jarník type theorems for the mixed affine forms and Jarník type theorem for the classical affine forms. We further prove that the sets of badly approximable affine forms, in both the classical and mixed settings, are hyperplane winning. The latter result, for the classical setting, answers a question raised by Kleinbock (1999).

AB - In this paper we discuss metric theory associated with the affine (inhomogeneous) linear forms in the so called doubly metric settings within the classical and the mixed setups. We consider the system of affine forms given by q↦qX+α, where q∈Zm (viewed as a row vector), X is an m×n real matrix and α∈Rn. The classical setting refers to the dist(qX+α,Zm) to measure the closeness of the integer values of the system (X,α) to integers. The absolute value setting is obtained by replacing dist(qX+α,Zm) with dist(qX+α,0); and the more general mixed settings are obtained by replacing dist(qX+α,Zm) with dist(qX+α,Λ), where Λ is a subgroup of Zm. We prove the Khintchine–Groshev and Jarník type theorems for the mixed affine forms and Jarník type theorem for the classical affine forms. We further prove that the sets of badly approximable affine forms, in both the classical and mixed settings, are hyperplane winning. The latter result, for the classical setting, answers a question raised by Kleinbock (1999).

KW - Hausdorff dimension

KW - Hausdorff measure

KW - Hyperplane winning

KW - Jarnik theorem

KW - Khintchine–Groshev theorem

KW - Schmidt game

UR - http://www.scopus.com/inward/record.url?scp=85076841487&partnerID=8YFLogxK

U2 - 10.1016/j.jnt.2019.11.014

DO - 10.1016/j.jnt.2019.11.014

M3 - Journal article

AN - SCOPUS:85076841487

SN - 0022-314X

VL - 213

SP - 67

EP - 100

JO - Journal of Number Theory

JF - Journal of Number Theory

ER -