Positive dyadic density for rational weighted binary expansions

Let $S\subseteq \mathbb{N}$ be infinite, and consider the weighted binary expansion ${\eta }_S={\sum }_{n\in S}n{2}^{-n}$. We prove that if ${\eta }_S$ is rational, then $S$ occupies a positive proportion of every sufficiently large dyadic block. The proportion depends only on the reduced denominator of ${\eta }_S$. This blockwise density theorem implies Erdős Problem~260. Rationality turns the scaled binary tails into nonnegative integer carries bounded linearly in the scale. If a dyadic block were sparse, its overlapping gap windows would give a positive lower bound for the integrated excess over a threshold interval. A lattice congruence places the carry states associated with each frequent initial long prefix on an affine occurrence line. Their continuations trace a path of normalized slopes. Odd denominators and separation of reduced fractions control paths that remain in $(0,1)$. Once a path leaves $[0,1]$, its distance from that interval grows exponentially, while the original integer parameter is retained for reconstruction. A partition into four disjoint classes combines the complementary multiplicity estimates into an upper bound for the same integrated excess. The two bounds contradict one another. Thus rationality forces a local density condition on every infinite support of this form.