Ramanujan Journal. An International Journal Devoted to the Areas of Mathematics Influenced by Ramanujan (Ramanujan J.) (20220101), 59, no.~4, 1007-1021. ISSN: 1382-4090 (print).eISSN: 1572-9303.
For an irrational number $\alpha > 1$, let $p_{\alpha}(n)$ and $q_{\alpha}(n)$ denote the number of partitions of $n$ into summands and distinct summands, respectively, chosen from the Beatty sequence $(\lfloor \alpha m\rfloor)_{m \geq 1}$. The irrationality measure $\mu(\alpha)$ of a real number $\alpha$ is defined by $$ \mu(\alpha)=\inf \{\mu \in \Bbb{R}: 0 < q^{-1} \Vert q\alpha \Vert < q^{-\mu} \ \text{has finitely many solutions} \ q \in \Bbb{N} \}, $$ where $\Vert x \Vert= \min_{n \in \Bbb{Z}}|x-n|$. If $\mu(\alpha)= \infty$, then $\alpha$ is called a Liouville number. \par In 1977, P. Erdős and L.~B. Richmond [in {\it Proceedings of the Seventh Manitoba Conference on Numerical Mathematics and Computing (Univ. Manitoba, Winnipeg, Man., 1977)}, pp. 371--377, Congress. Numer., XX, Utilitas Math., Winnipeg, MB, 1978; MR0535018] proved asymptotic formulas with error terms for $p_{\alpha}(n)$ and $q_{\alpha}(n)$ when the irrational number $\alpha > 1$ has finite irrationality measure $\mu(\alpha)$. But, for a Liouville number $\alpha > 1$, they could only explore the asymptotic behavior of $\log p_{\alpha}(n)$ and $\log q_{\alpha}(n)$. \par In this paper, the author establishes asymptotic formulas for $p_{\alpha}(n)$ and $q_{\alpha}(n)$ for every irrational number $\alpha > 1$, which improves some results of Erdős and Richmond. In particular, these asymptotic formulas for $p_{\alpha}(n)$ and $q_{\alpha}(n)$ are new for Liouville numbers $\alpha >1$.