An Almost-Covering Threshold for Golomb-Ruler Difference Packings

Abstract

For a fixed integer $t\geq 3$, 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: \[ P_t(U)=U-o(U) \quad\Longleftrightarrow\quad 3\leq t\leq 5. \] 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 $x_0\in(\pi,3\pi/2)$ is the first positive solution of $\tan x=x$ and \[ \gamma_0=-\frac{2\sin x_0}{x_0}=0.4344672564\ldots, \] then, for every fixed $t\geq 6$, \[ \liminf_{U\to\infty}\left(1-\frac{P_t(U)}{U}\right) \geq \frac{(t-1)\gamma_0-2}{2(t-2)}. \] In particular, the forced gap for six-mark rulers is at least $2.1542035\%$. We also prove a discrete small-difference bound which yields a stronger obstruction for every $t\geq14$ and forces a gap of \[ \frac12-\frac1{\sqrt t}-\frac7{8t}+O(t^{-3/2}) \] as $t\to\infty$.