Bernstein-Walsh inequalities in higher dimensions over exponential\n curves

2014/12/04 by Shirali Kadyrov, Kadyrov, Shirali, Mark G. Lawrence +2
Mathematics · #Meromorphic and Entire Functions #Advanced Differential Equations and Dynamical Systems #Analytic and geometric function theory

paper · pdf · doi:10.48550/arxiv.1412.1668

Abstract

Let bf x=(x₁,\…,x_d) \∈ [-1,1]^d be linearly independent over\n mathbb Z, set K= (e^z,e^{x₁ z},e^{x₂ z}\…,e^{x_d z}): |z| \≤ 1 .\nWe prove sharp estimates for the growth of a polynomial of degree n, in terms\nof E_n(
bf x):=
sup

|P
|
_Delta^d+1:P
in
mathcal P_n(d+1),
|P
|_K\n
le 1
, where \Δ^d+1 is the unit polydisk. For all bf x \∈\n[-1,1]^d with linearly independent entries, we have the lower estimate
log\nE_n(
bf x)
ge
fracn^d+1(d-1)!(d+1)
log n - O(n^d+1); for\nDiophantine bf x, we have
log E_n(
bf x)
le
frac\nn^d+1(d-1)!(d+1)
log n+O( n^d+1). In particular, this estimate holds\nfor almost all bf x with respect to Lebesgue measure.\n

Related

• Polynomial estimates over exponential curves in \mathbb C²
• Polynomial estimates, exponential curves and Diophantine approximation
• Quantitative bounds in the nonlinear Roth theorem
• Measures of polynomial growth and classical convolution inequalities
• Polynomial Szemerédi for sets with large Hausdorff dimension on the Torus