Partitions into Beatty sequences

Abstract

Let $\alpha >1$ be an irrational number. We establish asymptotic formulas for the number of partitions of n into summands and distinct summands, chosen from the Beatty sequence $(\lfloor \alpha m\rfloor )$. This improves some results of Erdös and Richmond established in 1977.

Appendix A: Numerical approximation for \(\Lambda _{\alpha }\)

In this appendix, we investigate the numerical approximation for $\Lambda _{\alpha }$. We define that

$$\begin{aligned} \Pi _{\alpha }=\prod _{\ell \ge 1}\left( 1-\frac{\{\alpha \ell \}}{\alpha \ell }\right) e^{\frac{1}{2\alpha \ell }}. \end{aligned}$$

(A.1)

Then the use of (1.5) implies that $\Lambda _{\alpha }=4\sqrt{3}(\pi e^{-\gamma })^{1/2\alpha }(\alpha /6)^{1/4\alpha }\Pi _{\alpha }$. Taking the logarithm of (A.1) we obtain

$$\begin{aligned} \log \Pi _{\alpha }&=\sum _{\ell \ge 1}\left( \frac{1}{2\alpha \ell }+\log \left( 1-\frac{\{\alpha \ell \}}{\alpha \ell }\right) \right) \nonumber \\&=\bigg (\sum _{1\le \ell \le N}+\sum _{\ell >N}\bigg )\left( \frac{1}{2\alpha \ell }+\log \left( 1-\frac{\{\alpha \ell \}}{\alpha \ell }\right) \right) \nonumber \\&=:\Sigma _m(N)+\Sigma _e(N), \end{aligned}$$

(A.2)

where the integer $N>10$ will be chosen for giving a good numerical approximation for $\Lambda _{\alpha }$.

We now bound the error term $\Sigma _e(N)$. We rewrite the sum of $\Sigma _e(N)$ as

$$\begin{aligned} \Sigma _{e}(N)&=-\sum _{\ell>N}\frac{\widetilde{B}_1(\alpha \ell )}{\alpha \ell }+\sum _{\ell >N}\left( \log \left( 1-\frac{\{\alpha \ell \}}{\alpha \ell }\right) +\frac{\{\alpha \ell \}}{\alpha \ell }\right) \nonumber \\&=\Sigma _{e1}(N)+\Sigma _{e2}(N). \end{aligned}$$

(A.3)

It is not difficult to give a bound for the second sum above that

$$\begin{aligned} 0<-\Sigma _{e2}(N)&< -\sum _{\ell >N}\left( \log \left( 1-\frac{1}{\alpha \ell }\right) +\frac{1}{\alpha \ell }\right) \nonumber \\&<-\int _{N}^{\infty }\left( \log \left( 1-\frac{1}{\alpha x}\right) +\frac{1}{\alpha x}\right) \,dx<\frac{3\alpha N-2}{6\alpha N(\alpha N-1)}. \end{aligned}$$

(A.4)

Using part integration to $\Sigma _{e1}(N)$ we have

$$\begin{aligned} \Sigma _{e1}(N)&=-\int _{N}^{\infty }\frac{1}{\alpha x}\,d S_{\alpha }(x)\nonumber \\&=\frac{S_{\alpha }(N)}{\alpha N}-\frac{1}{\alpha }\int _{N}^{\infty }\frac{S_{\alpha }(x)}{x^2}\,dx. \end{aligned}$$

(A.5)

We now focus on the approximation of $\Lambda _{\alpha }$ for a class of irrational numbers $\alpha$ in which the partial quotients of the continued fraction expansion of $\alpha$ are bounded. In other words, $\alpha$ has the following continued fraction expansion

$$\begin{aligned} \alpha =[a_0; a_1, a_2,\ldots ]=a_0+\frac{1}{a_1+\frac{1}{a_2+\cdots }} \end{aligned}$$

with all $a_j\le A$ for some $A>0$. In this case, Ostrowski [6, pp. 80–81] proved that

$$\begin{aligned} |S_{\alpha }(x)|\le \frac{3}{2}A\log x, \end{aligned}$$

(A.6)

for all $x>10$. Substituting (A.6) in (A.5) we find that

$$\begin{aligned} |\Sigma _{e1}(N)|\le \frac{3A\log N}{\alpha N}+\frac{3A}{2\alpha N}. \end{aligned}$$

Combining (A.2)–(A.4) and the above upper bound, we obtain

$$\begin{aligned} \left| \log \Pi _{\alpha }-\Sigma _m(N)\right| < \frac{3A}{\alpha N}\left( \log N+\frac{1}{2}\right) +\frac{3\alpha N-2}{6\alpha N(\alpha N-1)}. \end{aligned}$$

(A.7)

We now give the numerical approximation for $\Lambda _{\sqrt{2}}$. Note that $\sqrt{2}=[1;2,2,2,\ldots ]$, that is the partial quotients of the continued fraction expansion are bounded by 2. Therefore, in (A.7), the constant A equals 2 for $\alpha =\sqrt{2}$. Hence

$$\begin{aligned} \left| \log \Pi _{\sqrt{2}}-\Sigma _m(N)\right| < \frac{3\sqrt{2}}{N}\left( \log N+\frac{1}{2}\right) +\frac{3N-\sqrt{2}}{6N(\sqrt{2} N-1)}. \end{aligned}$$

Taking $N=10^6$, then using Mathematica we find that

$$\log \Pi _{\sqrt{2}}=-0.127496+ 6.11\times 10^{-5}\theta ,$$

for some $\theta \in (-1,1)$. Hence

$$\begin{aligned} \Lambda _{\sqrt{2}}=4\sqrt{3}(\pi e^{-\gamma })^{1/2\sqrt{2}}(\sqrt{2}/6)^{1/4\sqrt{2}}\Pi _{\sqrt{2}}\in (5.7731, 5.7739). \end{aligned}$$