Badly approximable systems of linear forms

AbstractA number α is called badly approximable if there is a constant c = c(α)>0 such that |q|αq − p| > c(α) for any integers p, q ≠ 0. The existence of continuum—many badly approximable numbers follows easily from the theory of continued fractions. In the present paper the notion of a badly approximable number is generalized to badly approximable systems of m linear forms in n variables. Among other results we prove for every given m and n the existence of continuum many badly approximable systems of linear forms