Sufficient conditions for the infinitely-recurring difference set $D(A)$ to have bounded gaps (Erdős #332)

Let $A\subseteq\mathbb{N}$. Define $D(A)$ to be the set of all positive integers $d$ that occur infinitely often as a difference $a_1-a_2$ with $a_1,a_2\in A$ (that is, $d$ for which $a_1-a_2=d$ has infinitely many solutions in $A$). Find natural conditions on $A$ sufficient to guarantee that $D(A)$ has bounded gaps, i.e. that there is a constant $C=C(A)$ such that every interval of length $C$ contains an element of $D(A)$ (equivalently, consecutive elements of $D(A)$ differ by at most $C$). One may also seek conditions ensuring the weaker conclusions that $D(A)$ has positive density, that $\sum_{d\in D(A)}\frac1d=\infty$, or merely that $D(A)\neq\emptyset$.

Acceptance. FULLY RESOLVES: a complete proof characterizing the conditions on $A$ that force $D(A)$ to have bounded gaps (a sharp sufficient condition together with a matching necessity example) — machine-checkable in Lean/Coq preferred, otherwise a full written proof with all steps. ADVANCES (each independently checkable): (a) prove a sufficient condition strictly weaker than positive density — an explicit density-zero or structural hypothesis $H$, with a proof that $H$ implies $D(A)$ has bounded gaps and an example set satisfying $H$ but not having positive density; or (b) under a stated hypothesis weaker than positive density, prove one of the weaker conclusions (positive density of $D(A)$, $\sum_{d\in D(A)}1/d=\infty$, or $D(A)\neq\emptyset$); or (c) construct a counterexample delimiting what is impossible. Any claimed sufficient condition must be strictly weaker than the positive-density hypothesis stated in the background. Deliver a proof file (Lean preferred) or a complete written proof.