OlympiadBench / 1920 / For any two different real numbers x and y, we define D(x,y) to be the…

For any two different real numbers x and y, we define D(x,y) to be the unique integer d satisfying 2^(d)≤|x−y|<2^(d+1). Given a set of reals F, and an element x∈F, we say that the scales of x in F are the values of D(x,y) for y∈F with x≠y.

Let k be a given positive integer. Suppose that each member x of F has at most k different scales in F (note that these scales may depend on x ). What is the maximum possible size of F ?

For any two different real numbers $x$ and $y$, we define $D(x, y)$ to be the unique integer $d$ satisfying $2^{d} \leqslant|x-y|<2^{d+1}$. Given a set of reals $\mathcal{F}$, and an element $x \in \mathcal{F}$, we say that the scales of $x$ in $\mathcal{F}$ 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 $x$ of $\mathcal{F}$ has at most $k$ different scales in $\mathcal{F}$ (note that these scales may depend on $x$ ). What is the maximum possible size of $\mathcal{F}$ ?