MathDB

The sequence

c_{0}, c_{1}, . . . , c_{n}, . . .

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=\textstyle\sum_{j \in J}{c_{j}}

y=\textstyle\sum_{j \in J}{c_{j-1}}

. Prove that there exist real numbers

\alpha

\beta

, and

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

.
Remark: A sum over the elements of the empty set is assumed to be

0

Problem Statement