A multidimensional version of Turán's lemma

In this article we provide a multidimensional version of Nazarov'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 we let Tn≔T×⋯×T represent the distinguished boundary of the polydisk Dn≔D×⋯×D for n∈N and D the open unit disk then, as in the one dimensional case, the constant which relates the uniform norm of p on Tn to the uniform norm of p on any measurable subset E of Tn depends on the order of p and the measure of the set E.