An Almost-Covering Threshold for Golomb-Ruler Difference Packings

For a fixed integer $t\ge 3$, consider families of $t$-mark Golomb rulers whose positive-difference sets are pairwise disjoint and contained in $[1,U]$. Let $P_t(U)$ be the largest number of integers covered by such a family. We determine the threshold for asymptotically complete coverage: $$P_t(U)=U-o(U)⟺3\le t\le 5.$$ The cases $t=3,4$ follow from the known existence spectra for perfect difference families. For $t=5$, Wild’s product construction, in the form recorded by Mathon and applied to perfect families of orders $121$ and $161$, gives a multiplicative semigroup of exact-covering scales; an elementary density lemma on its logarithms then supplies a scale $(1-o(1))U$ below every sufficiently large $U$. For the converse, we give a self-contained one-frequency Fourier obstruction. If $x_0\in (\pi ,3\pi \mathrm{/}2)$ is the first positive solution of $\tan x=x$ and $${\gamma }_0=-\frac{2\sin x_0}{x_0}=0.4344672564\dots ,$$ then, for every fixed $t\ge 6$, $$\underset{U\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}(1-\frac{P_t(U)}{U})\ge \frac{(t-1){\gamma }_0-2}{2(t-2)}\mathrm{.}$$ In particular, the forced gap for six-mark rulers is at least $2.1542035\mathrm{\%}$. We also prove a discrete small-difference bound which yields a stronger obstruction for every $t\ge 14$ and forces a gap of $$\frac{1}{2}-\frac{1}{\sqrt{t}}-\frac{7}{8t}+O(t^{-3\mathrm{/}2})$$ as $t\to \mathrm{\infty }$.