### Abstract

Let x= (x_{1}, ⋯, x_{d}) ∈ [- 1 , 1] ^{d} be linearly independent over Z, and set K={(ez,ex1z,ex2z⋯,exdz):|z|≤1}. We prove sharp estimates for the growth of a polynomial of degree n, in terms of En(x):=sup{‖P‖Δd+1:P∈Pn(d+1),‖P‖K≤1},where Δ ^{d}^{+}^{1} is the unit polydisk. For all x∈ [- 1 , 1] ^{d} with linearly independent entries, we have the lower estimate logEn(x)≥nd+1(d-1)!(d+1)logn-O(nd+1);for Diophantine x, we have logEn(x)≤nd+1(d-1)!(d+1)logn+O(nd+1).In particular, this estimate holds for almost all x with respect to Lebesgue measure. The results here generalize those of Coman and Poletsky [6] for d= 1 without relying on estimates for best approximants of rational numbers which do not hold in the vector-valued setting.

Original language | English |
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Pages (from-to) | 327-338 |

Number of pages | 12 |

Journal | Constructive Approximation |

Volume | 44 |

Issue number | 3 |

DOIs | |

Publication status | Published - Dec 1 2016 |

### Keywords

- Bernstein–Walsh inequalities
- Hausdorff dimension
- Liouville vectors
- Polynomial inequalities

### ASJC Scopus subject areas

- Analysis
- Mathematics(all)
- Computational Mathematics

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## Cite this

*Constructive Approximation*,

*44*(3), 327-338. https://doi.org/10.1007/s00365-015-9314-2