Bernstein-walsh inequalities in higherdimensions over exponential curves

Abstract

Let x = (x1; : : : ; xd) 2 [􀀀1; 1]d be linearly independent over Z, set K = f(ez; ex1z; ex2z : : : ; exdz) : jzj 1g:We prove sharp estimates for the growth of a polynomial of degree n, in terms of En(x) := supfkPk d+1 : P 2 Pn(d + 1); kPkK 1g; where d+1 is the unit polydisk. For all x 2 [􀀀1; 1]d with linearly independent entries, we have the lower estimate logEn(x) nd+1 (d 􀀀 1)!(d + 1) log n 􀀀 O(nd+1); for Diophantine x, we have logEn(x) nd+1 (d 􀀀 1)!(d + 1) log n + O(nd+1): In particular, this estimate holds for almost all x with respect to Lebesgue measure. The results here generalize those of [6] for d = 1, without relying on estimates for best approximants of rational numbers which do not hold in the vector-valued setting.