Short interval results for a class of integers
Integers whose prime factorizations are subject to various constraints have long been a subject of fascination for number theorists. In particular, a great deal of effort has been dedicated to understanding the distribution of such integers. In this paper, the author studies the distribution of ``\(\mathcal G\)-numbers'' in short intervals. Suppose that \(\mathcal G\) is a set of positive integers which has a finite subset \(\{ g_1, \dots, g_s \}\) with \(\gcd(g_1, \dots, g_s) = 1\). We call a positive integer \(n\) a \textit{\(\mathcal G\)-number}, if all the exponents \(a_1, \dots, a_s\) in its prime factorization \(n = p_1^{a_1}p_2^{a_2} \cdots p_s^{a_s}\) belong to \(\mathcal G\). For example, when \(\mathcal G = \{1, 2, \dots, k - 1\}\), the \(\mathcal G\)-numbers are the \(k\)-free integers; and when \(\mathcal G = \{k, k + 1, \dots \}\), the \(\mathcal G\)-numbers are the \(k\)-full integers. With \(\mathcal G\) as above, we write \(g = \min \mathcal G\) and \(\delta(n; \mathcal G)\) for the characteristic function of the \(\mathcal G\)-numbers. The paper studies the range of \(y\) for which one has an asymptotic formula of the form \[ \sum_{x < n \leq x + y} \delta(n; \mathcal G) = yx^{-\alpha}(c + o(1)) \qquad (x \to \infty); \tag{1} \] here \(\alpha = 1 - g^{-1}\) and \(c = c(\mathcal G)\) is a constant depending only on the set \(\mathcal G\). It is easy to show that one can take \(y = x^{\theta}\) for any fixed \(\theta > \alpha + (g + 1)^{-1}\), and the primary goal is to relax the requirement for \(\theta\). The author shows that such results on the distribution of \(\mathcal G\)-numbers can be derived from results on two classical lattice point problems: the problem of estimating the number of lattice points close to a smooth plane curve; and the problem of estimating the remainders in asymptotic formulas of the form \[ \mathop{\sum \; 1}_{n_1^{u_1} \cdots n_r^{u_r} \leq x} = H(x; u_1, \dots, u_r) + O \left( x^{\beta} \right), \tag{2} \] where \(H(x; u_1, \dots, u_r)\) is a (smooth) main term and \(\beta = \beta(u_1, \dots, u_r)\) is an explicit exponent. Since most of the paper's results are too technical to state here, we provide a representative sample of special cases: When \(1 \in \mathcal G\), it is shown that (1) holds with \(y = x^{\theta}\) for any fixed \(\theta > (2k + 1)^{-1}\), where \(k\) is the least integer with \(k \notin \mathcal G\). This is a consequence from an estimate for the number of lattice points close to a smooth curve due to \textit{M. Filaseta} and \textit{O. Trifonov} [Proc. Lond. Math. Soc. (3) 73, No.~2, 241--278 (1996; Zbl 0867.11053)]. Another result states that when \(1 \notin \mathcal G\) and \(2, 3 \in \mathcal G\), then (1) holds with \(y = x^{\theta}\) for any fixed \(\theta > 5/8\); this is derived from another estimate for the number of lattice points close to a smooth curve by \textit{M. N. Huxley} and \textit{O. Trifonov} [Math. Proc. Camb. Philos. Soc. 119, No.~2, 201--208 (1996; Zbl 0852.11046)]. Information of type (2) is needed when \(g\) is larger. For example, if \(1, 2, 3 \notin \mathcal G\) and \(4, 5, 6, 7 \in \mathcal G\), a result of \textit{P. Sargos} and \textit{J. Wu} [Acta Math. Hung. 87, No.~4, 333--354 (2000; Zbl 0963.11045)] yields (1) with \(y = x^{\theta}\) for any fixed \(\theta > \alpha + 1/10 = 0.85\).
scientific article; zbMATH DE number 2078307