On regular systems of real algebraic numbers of third degree in short intervals (Q483179)

{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 \textit{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 \[ \begin{split} & N(\gamma_i)\leq T,\;1\leq i\leq t,\\ &|\gamma_i-\gamma_j|>T^{-1},\;1\leq i<j\leq t,\\ &t>c_1|I|T. \end{split} \] \textit{A. Baker} and \textit{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 \textit{V. I. Bernik} [Acta Arith. 53, No. 1, 17--28 (1989; Zbl 0692.10042)]. \textit{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.