Variation of a Beatty Sequence

Question

I've been studying Beatty sequences lately, and, having read and understood a proof of Raleigh's theorem, I know that if $\lfloor \alpha x\rfloor$ is a Beatty sequence with $\alpha\gt 1$, then its complementary Beatty sequence is $\lfloor \beta x\rfloor$, where $$\frac{1}{\alpha}+\frac{1}{\beta}=1$$ However, I'm having a little bit of trouble with the following: as a generalization of the Beatty sequence $\lfloor \phi n\rfloor$, I am considering the similar sequence $\lfloor \phi n-1/3\rfloor$ and trying to find its complement in the natural numbers. I believe that its complement is $\lfloor (\phi+1)n+2/3\rfloor$, but the method of proof used for Beatty's theorem doesn't work in this case. I used the second proof provided here.

EDIT: My conjecture is false (see comments), but I would still like to find the complement of the originally mentioned sequence.

Does anyone know how to find the complement of the following sequence? $$\{\lfloor \phi n-1/3\rfloor\}_{n=1}^\infty$$

Answer 1

To make the Wikipedia proof work, the shifts have to cancel. If we denote your shift of $\frac13$ by $\delta$ and the unknown shift for the complementary sequence by $\epsilon$, the inequalities in the proof become

$$ j < k \cdot r - \delta < j + 1 \text{ and } j < m \cdot s + \epsilon < j + 1\;, $$

and then dividing through by $r$ and $s$, respectively, and adding as in the ordinary proof yields

$$ j < k -\frac\delta r + m +\frac\epsilon s< j + 1\;. $$

Thus for the proof to go through, you need $\frac\epsilon s-\frac\delta r\in\mathbb Z$. Thus, given $\delta$, you need to choose

$$\epsilon=ks+\delta\cdot\frac sr\;\text{ with }\;k\in\mathbb Z\;.$$

In your case, with $\delta\cdot\frac sr=\frac\phi3$, the simplest choice is $\epsilon=\frac\phi3$ to get the complementary sequence $\lfloor (\phi+1)n+\phi/3\rfloor$.

Answer 2

draw a line $y = kx + r$ for $r \in (0;1)$ and $k > 0$ that doesn't go through any point with integer coordinates.

Starting from $(0,r)$, that line will intersect the horizontal lines $x=n$ and vertical lines $y=n$ one at a time (it can never cross two at once since I explicitly said so)

By giving a number to each crossing starting with $0$ for the vertical line $x=0$, you can define two functions $h(n)$ and $v(n)$ such that $h(n)$ is the number of the crossing that crosses the line $y=n$ and similarly for $h(n)$ and horizontal lines.

Now, what is $v(n)$ ? It is $n$ plus the number of horizontal lines that were crossed while reaching the vertical line $x=n$. Putting $x=n$ and solving for $y$ you get that the line $x=n$ is crossed at $(x = n, y=kn+r)$. Hence, $v(n) = n + \lfloor kn + r \rfloor = \lfloor (k+1)n+r \rfloor$

Similarly for $h(n)$, putting $y=n$ and solving for $x$ you get the point $(x = \frac{n-r}k, y=n)$, and so $h(n) = n + \lfloor\frac {n-r}k \rfloor= \lfloor \frac {(k+1)n-r}k \rfloor$

Now by construction the sequences $(h(n))_{n \ge 1}$ and $(v(n))_{n \ge 1}$ are a partition of $\{1;2;3;\cdots\}$

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For your problem you would like to have $(k+1)/k = \phi$ and $r/k = 1/3$, so $k = \phi$ and $r = \phi/3$.

The line is $y = \phi x + (\phi/3)$, or $y/\phi = x + 1/3$, and it can't go through any integer point (or else $\phi$ would be rational) so the preceding section applies :

$h(n) = \lfloor \phi n - \frac 13 \rfloor$ and $v(n) = \lfloor (\phi+1)n + \phi/3 \rfloor$ give a partition of $\Bbb N$.

So your answer is $\lfloor (\phi+1)n + \phi/3 \rfloor$ instead of $\lfloor (\phi+1)n + 2/3 \rfloor$