Abstract
In the theory of Diophantine Approximations one considers a question on approximation of Real Numbers by rational fractions with one and the same denominators. Among intensively studied questions of this theory a special place occupy Metric questions. Here such questions of the theory are considered which take place for almost all real numbers from given interval. For the first time similar questions have been studied by Khintchine for approximation of independent quantities. It had been investigated by him conditions at which for almost all real numbers specified accuracy of approximation is reached. Very important in the technical plan Khintchince’s transference principle allows us to connect a simultaneous approximations of dependent quantities with approximations of linear forms with integral coefficients. In 1932 Mahler K. has entered classification of transcendental numbers into consideration. He showed that almost all transcendental numbers are 𝑆 -numbers. Moreover, Mahler had proved an existence of a constant 𝛾 > 0 such that for almost all 𝜔 $$|𝑃(𝜔)| > ℎ^(−𝑛𝛾)$$, for all polynomials 𝑃 of degree no more 𝑛 and height ℎ > ℎ0(𝜔, 𝑛, 𝛾). Mahler showed that it is possible to take $$𝛾 = 4 + 𝜀$$. In the same work Mahler made an assumption that it is possible to take 𝛾 = 1 + 𝜀 for almost all real numbers. This hypothesis was proved by Sprindzhuk V. G by a method of essential and nonessential domains. Simultaneously, Sprindzhuk V. G. advanced some hypotheses generalising and improving Mahler’s results. Further these investigations of Sprindzuk led to the development of a new direction in the theory of Diophantine Approximations – to the research of extremality of manifolds. In the present article we develop a new approach to these questions and offer a new proof for extremality of algebraic varieties. This method allows to establish extremality of affine image of topological product of some varieties. Considering one particular case, we prove that the extremality for these varieties is possible to deduct from theorems on the convergence exponent of a special integral of Terry’s problem, using E.I. Kovalevskaya’s lemma. Further, we derive from the getting result particular case of the Sprindzuk’s hypothesis on extremality of varieties, induced by monomials of a polynomial in two variables.
Highlights
This hypothesis was proved by Sprindzhuk V
G. advanced some hypotheses generalising and improving Mahler’s results. Further these investigations of Sprindzuk led to the development of a new direction in the theory of Diophantine Approximations – to the research of extremality of manifolds
Considering one particular case, we prove that the extremality for these varieties is possible to deduct from theorems on the convergence exponent of a special integral of Terry’s problem, using E.I
Summary
Для установления наших результатов перейдем к равносильному варианту поставленной задачи, выражая ее в виде экстремальности многообразий. Что полученные выше соотношения можно интерпретировать по-другому. Обозначим через v(η1, η2, ..., ηn) точную верхнюю грань всех тех v > 0, для которых (2) выполнено для бесконечного числа целых векторов (b1, ..., bn). Одновременно, рассмотрим следующее неравенство: max (‖qη1‖ , ‖qη2‖ , ..., ‖qηn‖) q−u. Теперь обозначим через u(η1, η2, ..., ηn) точную верхнюю грань всех тех u > 0, для которых (3) выполнено для бесконечного числа целых положительных q > 0. [3, 7, 10]), утверждающая, что равенства v(η1, η2, ..., ηn) = n и u(η1, η2, ..., ηn) = 1/n равносильны. Полагая n = 9 в качестве чисел η1, η2, ..., η9 возьмем следующие многочлены в топологическом произведении многообразий, порожденных одночленами положительной степени рассматриваемого многочлена P :
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