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On regular systems of real algebraic numbers of third degree in short intervals. (English. Russian original) Zbl 1304.11065
Proc. Steklov Inst. Math. 282, Suppl. 1, S54-S66 (2013); translation from Sovrem. Probl. Mat. 17, 61-75 (2013).
Definitions: Let \(A_n\) be the set of real algebraic numbers of degree \(n\). The height \(H(\alpha)\) of an algebraic number \(\alpha\) is the maximum of the absolute values of the coefficients of the minimal polynomial of \(\alpha\). In the sequel, \(c_i\) denotes a constant for the index \(i\).
Let \(\Gamma\) be a countable set of real numbers and \(N : \Gamma\rightarrow \mathbb R\) be a positive function. The pair \((\Gamma,N)\) is called a regular system if there exists a constant \(c_1 = c_1(\Gamma,N) > 0\), such that for any interval \(I \subset \mathbb R\) there exists a sufficiently large number \(T_0 = T_0(\Gamma,N, I) > 0 \), such that for any integer \(T > T_0\) there exist \(\gamma_1, \gamma_2,\dots,\gamma_t\) in \(\Gamma\cap I\) such that \[ \begin{split} & N(\gamma_i)\leq T,\;1\leq i\leq t,\\ &|\gamma_i-\gamma_j|>T^{-1},\;1\leq i<j\leq t,\\ &t>c_1|I|T. \end{split} \] A. Baker and W. M. Schmidt introduced the definition of regular systems in [Proc. Lond. Math. Soc. (3) 21, 1–11 (1970; Zbl 0206.05801)] and proved the regularity of real algebraic numbers of any degree. This gives the possibility of obtaining the lower bound for the Hausdorff dimension of the set of real numbers which are approximated by algebraic numbers with a given order of approximation. The result of Baker and Schmidt was improved by V. I. Bernik [Acta Arith. 53, No. 1, 17–28 (1989; Zbl 0692.10042)]. V. Beresnevich [Acta Arith. 90, No. 2, 97–112 (1999; Zbl 0937.11027)] proved the regularity of real algebraic numbers with the best posible result for the functions defined on the set of algebraic numbers. As an example, the set of rational numbers \(p/q\), \(\mathrm{gcd}(p, q) = 1\), \(N(p/q) = q^2\) is a regular system. As another example of the above results, it was shown that the set of real algebraic numbers \(\alpha\) of degree \(n\) together with the function \(N(\alpha) = H(\alpha)^{n+1}\log^{-v} H(\alpha)\) forms a regular system for \(v = 3n(n + 1)\), \(2\) and \(0\) respectively.
Results: The authors prove the following result (Theorem 1). Let \(I\) be a finite interval contained in \([-1/2, 1/2]\). Then there exist positive constants \(c_1, c_4\) and a positive number \(T_0 = c_4|I|^{-4}\) such that for any \(T \geq T_0\) there exist numbers \(\alpha_1, \dots, \alpha_t \in A_3\cap I\) such that \[ \begin{split} & H(\alpha_i) \leq T^{1/4}, \;1 \leq i \leq t,\\ & |\alpha_i - \alpha_j| \geq T^{-1},\;1 \leq i < j \leq t,\\ & t \geq c_1|I|T.\\ \end{split} \] This implies that the set of real algebraic numbers \(\alpha\) of degree \(3\), together with the function \(N(\alpha) = H^4(\alpha)\) form a regular system on \([-1/2, 1/2]\). The authors provide also a metric theorem used for the proof of Theorem 1.
MSC:
11J17 Approximation by numbers from a fixed field
11J68 Approximation to algebraic numbers
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References:
[1] A. Baker and W. M. Schmidt, ”Diophantine approximation and Hausdorff dimension,” Proc. Lond. Math. Soc. (3) 21, 1–11 (1970). · Zbl 0206.05801 · doi:10.1112/plms/s3-21.1.1
[2] V. I. Bernik, ”On the exact order of approximation of zero by values of integral polynomials,” Acta Arith. 53(1), 17–28 (1989). · Zbl 0692.10042 · doi:10.4064/aa-53-1-17-28
[3] V. Beresnevich, ”On approximation of real numbers by real algebraic numbers,” Acta Arith. 90(2), 97–112 (1999). · Zbl 0937.11027 · doi:10.4064/aa-90-2-97-112
[4] Y. Bugeaud, Approximation by Algebraic Numbers (Cambridge Univ. Press, Cambridge, 2004). · Zbl 1055.11002
[5] V. V. Beresnevich, ”Effective measure estimates for sets of real numbers with a given error of approximation by quadratic irrationalities,” Izv. Akad. Nauk Belarusi, Ser. Fiz.-Mat. Nauk, No. 4, 10–15 (1996). · Zbl 0871.11049
[6] V. G. Sprindžuk,, Mahler’s Problem in the Metric Theory of Numbers (Nauka i Tekkhnika, Minsk, 1967; Am.Math. Soc., Providence, RI, 1969).
[7] V. I. Bernik, ”Application of Hausdorff dimension in the theory of Diophantine approximations,” (Am. Math. Soc., Providence, RI, 1988), pp. 15–44. · Zbl 0655.10051
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