MathDB

For any two different real numbers

x

and

y

, we define

D(x,y)

to be the unique integer

d

satisfying

2^d\le |x-y| < 2^{d+1}

. Given a set of reals

\mathcal F

, and an element

x\in \mathcal F

, we say that the scales of

x

\mathcal F

are the values of

D(x,y)

for

y\in\mathcal F

with

x\neq y

. Let

k

be a given positive integer. Suppose that each member

x

\mathcal F

has at most

k

different scales in

\mathcal F

(note that these scales may depend on

x

). What is the maximum possible size of

\mathcal F

C9