Subfield: Algebra


Question: The sequence $c_{0}, c_{1}, \ldots, c_{n}, \ldots$ is defined by $c_{0}=1, c_{1}=0$ and $c_{n+2}=c_{n+1}+c_{n}$ for $n \geq 0$. Consider the set $S$ of ordered pairs $(x, y)$ for which there is a finite set $J$ of positive integers such that $x=\sum_{j \in J} c_{j}, y=\sum_{j \in J} c_{j-1}$. Prove that there exist real numbers $\alpha, \beta$ and $m, M$ with the following property: An ordered pair of nonnegative integers $(x, y)$ satisfies the inequality

$$
m<\alpha x+\beta y<M
$$

if and only if $(x, y) \in S$.

N. B. A sum over the elements of the empty set is assumed to be 0 .

Solution:
Let $\varphi=(1+\sqrt{5}) / 2$ and $\psi=(1-\sqrt{5}) / 2$ be the roots of the quadratic equation $t^{2}-t-1=0$. So $\varphi \psi=-1, \varphi+\psi=1$ and $1+\psi=\psi^{2}$. An easy induction shows that the general term $c_{n}$ of the given sequence satisfies

$$
c_{n}=\frac{\varphi^{n-1}-\psi^{n-1}}{\varphi-\psi} \quad \text { for } n \geq 0
$$

Suppose that the numbers $\alpha$ and $\beta$ have the stated property, for appropriately chosen $m$ and $M$. Since $\left(c_{n}, c_{n-1}\right) \in S$ for each $n$, the expression

$$\alpha c_{n}+\beta c_{n-1}=\frac{\alpha}{\sqrt{5}}\left(\varphi^{n-1}-\psi^{n-1}\right)+\frac{\beta}{\sqrt{5}}\left(\varphi^{n-2}-\psi^{n-2}\right)=\frac{1}{\sqrt{5}}\left[(\alpha \varphi+\beta) \varphi^{n-2}-(\alpha \psi+\beta) \psi^{n-2}\right]$$
is bounded as $n$ grows to infinity. Because $\varphi>1$ and $-1<\psi<0$, this implies $\alpha \varphi+\beta=0$.

To satisfy $\alpha \varphi+\beta=0$, one can set for instance $\alpha=\psi, \beta=1$. We now find the required $m$ and $M$ for this choice of $\alpha$ and $\beta$.

Note first that the above displayed equation gives $c_{n} \psi+c_{n-1}=\psi^{n-1}, n \geq 1$. In the sequel, we denote the pairs in $S$ by $\left(a_{J}, b_{J}\right)$, where $J$ is a finite subset of the set $\mathbb{N}$ of positive integers and $a_{J}=\sum_{j \in J} c_{j}, b_{J}=\sum_{j \in J} c_{j-1}$. Since $\psi a_{J}+b_{J}=\sum_{j \in J}\left(c_{j} \psi+c_{j-1}\right)$, we obtain

$$
\psi a_{J}+b_{J}=\sum_{j \in J} \psi^{j-1} \quad \text { for each }\left(a_{J}, b_{J}\right) \in S \tag{1}
$$

On the other hand, in view of $-1<\psi<0$,

$$
-1=\frac{\psi}{1-\psi^{2}}=\sum_{j=0}^{\infty} \psi^{2 j+1}<\sum_{j \in J} \psi^{j-1}<\sum_{j=0}^{\infty} \psi^{2 j}=\frac{1}{1-\psi^{2}}=1-\psi=\varphi
$$

Therefore, according to (1),

$$
-1<\psi a_{J}+b_{J}<\varphi \quad \text { for each }\left(a_{J}, b_{J}\right) \in S
$$

Thus $m=-1$ and $M=\varphi$ is an appropriate choice.

Conversely, we prove that if an ordered pair of nonnegative integers $(x, y)$ satisfies the inequality $-1<\psi x+y<\varphi$ then $(x, y) \in S$.



Lemma. Let $x, y$ be nonnegative integers such that $-1<\psi x+y<\varphi$. Then there exists a subset $J$ of $\mathbb{N}$ such that

$$
\psi x+y=\sum_{j \in J} \psi^{j-1} \tag{2}
$$

Proof. For $x=y=0$ it suffices to choose the empty subset of $\mathbb{N}$ as $J$, so let at least one of $x, y$ be nonzero. There exist representations of $\psi x+y$ of the form

$$
\psi x+y=\psi^{i_{1}}+\cdots+\psi^{i_{k}}
$$

where $i_{1} \leq \cdots \leq i_{k}$ is a sequence of nonnegative integers, not necessarily distinct. For instance, we can take $x$ summands $\psi^{1}=\psi$ and $y$ summands $\psi^{0}=1$. Consider all such representations of minimum length $k$ and focus on the ones for which $i_{1}$ has the minimum possible value $j_{1}$. Among them, consider the representations where $i_{2}$ has the minimum possible value $j_{2}$. Upon choosing $j_{3}, \ldots, j_{k}$ analogously, we obtain a sequence $j_{1} \leq \cdots \leq j_{k}$ which clearly satisfies $\psi x+y=\sum_{r=1}^{k} \psi^{j_{r}}$. To prove the lemma, it suffices to show that $j_{1}, \ldots, j_{k}$ are pairwise distinct.

Suppose on the contrary that $j_{r}=j_{r+1}$ for some $r=1, \ldots, k-1$. Let us consider the case $j_{r} \geq 2$ first. Observing that $2 \psi^{2}=1+\psi^{3}$, we replace $j_{r}$ and $j_{r+1}$ by $j_{r}-2$ and $j_{r}+1$, respectively. Since

$$
\psi^{j_{r}}+\psi^{j_{r+1}}=2 \psi^{j_{r}}=\psi^{j_{r}-2}\left(1+\psi^{3}\right)=\psi^{j_{r}-2}+\psi^{j_{r}+1}
$$

the new sequence also represents $\psi x+y$ as needed, and the value of $i_{r}$ in it contradicts the minimum choice of $j_{r}$.

Let $j_{r}=j_{r+1}=0$. Then the sum $\psi x+y=\sum_{r=1}^{k} \psi^{j_{r}}$ contains at least two summands equal to $\psi^{0}=1$. On the other hand $j_{s} \neq 1$ for all $s$, because the equality $1+\psi=\psi^{2}$ implies that a representation of minimum length cannot contain consecutive $i_{r}$ 's. It follows that

$$
\psi x+y=\sum_{r=1}^{k} \psi^{j_{r}}>2+\psi^{3}+\psi^{5}+\psi^{7}+\cdots=2-\psi^{2}=\varphi
$$

contradicting the condition of the lemma.

Let $j_{r}=j_{r+1}=1$; then $\sum_{r=1}^{k} \psi^{j_{r}}$ contains at least two summands equal to $\psi^{1}=\psi$. Like in the case $j_{r}=j_{r+1}=0$, we also infer that $j_{s} \neq 0$ and $j_{s} \neq 2$ for all $s$. Therefore

$$
\psi x+y=\sum_{r=1}^{k} \psi^{j_{r}}<2 \psi+\psi^{4}+\psi^{6}+\psi^{8}+\cdots=2 \psi-\psi^{3}=-1
$$

which is a contradiction again. The conclusion follows.

Now let the ordered pair $(x, y)$ satisfy $-1<\psi x+y<\varphi$; hence the lemma applies to $(x, y)$. Let $J \subset \mathbb{N}$ be such that (2) holds. Comparing (1) and (2), we conclude that $\psi x+y=\psi a_{J}+b_{J}$. Now, $x, y, a_{J}$ and $b_{J}$ are integers, and $\psi$ is irrational. So the last equality implies $x=a_{J}$ and $y=b_{J}$. This shows that the numbers $\alpha=\psi, \beta=1, m=-1, M=\varphi$ meet the requirements.