A multidimensional version of Turán's Lemma (Q2493075)

The author provides a multidimensional version of Navarov's extension of Turán's lemma -- a result in which the uniform norm of a complex-valued polynomial \(p\), on the unit circle \(T\) is compared with the uniform norm of \(p\) on any measurable subset of \(T\). If \(T^n:= T\times T\times\cdots\times T\) represent the distinguished boundary of the polydisk \(D^n:= D\times D\times\cdots\times D\) for \(n\in\mathbb{N}\) and \(D\) the open unit disk then, as in the one-dimensional case, the constant which relates the uniform norm of \(p\) on \(T^n\) to the uniform norm of \(p\) on any measurable subset \(E\) of \(T^n\) depends on the order of \(p\) and the measure of the set \(E\).