Hegyvári’s theorem on complete sequences, II
Tom 203 / 2022
Streszczenie
Let be a sequence of nonnegative integers. A sequence A is said to be complete if every sufficiently large integer can be represented as a finite sum of distinct terms of A. For a sequence S=\{s_1,s_2,\ldots \} and a real number \alpha \gt 0, let S_{\alpha }=\{\lfloor \alpha s_1\rfloor ,\lfloor \alpha s_2\rfloor ,\ldots \}, U_S=\{\alpha :S_{\alpha } \text { is complete}\} and let \mu (U_S) be the Lebesgue measure of U_S. In 2013, Chen and Fang improved a 1995 result of Hegyvári by proving that for 1 \lt \gamma \lt 2, if s_{n+1} \lt \gamma s_n (n\ge n_0) and U_S\neq \emptyset , then \mu (U_S) \gt 0, and proved that U_S\neq \emptyset if 1 \lt \gamma \lt 7/4. Recently, Fang and Liu showed that U_S\neq \emptyset if 1 \lt \gamma \lt 1.898\dots . It is known that for any \gamma \gt 2, there exists a sequence S with s_n \lt s_{n+1} \lt \gamma s_n (n\ge n_0) such that U_S=\emptyset . In this paper, we prove that U_S\neq \emptyset if 1 \lt \gamma \lt 2. This gives an affirmative answer to a problem posed by Chen and Fang.