Fix an integer t≥ 3. For a t-element set A⊂Z, let Delta(A)=a_j-a_i:a_i,a_j∈ A, a_i<a_j be its set of positive differences. Call A a Golomb ruler if all elements of Delta(A) are distinct. For a family F of t-mark Golomb rulers whose difference sets are pairwise disjoint and satisfy bigcup_A∈FDelta(A)⊆[1,U]capZ, define P_t(U) to be the maximum possible value of |bigcup_A∈FDelta(A)|. The claimed threshold theorem is P_t(U)=U-o(U) as U→∞ if and only if 3≤ t≤ 5. For every fixed t≥ 6, the claimed obstruction is liminf_U→∞(1-fracP_t(U)U)≥frac(t-1)gamma_0-22(t-2), where x_0∈(pi,3 pi/2) is the first positive solution of tan x=x and gamma_0=-2 sin(x_0)/x_0=0.4344672564….

Almost-covering problem for Golomb-ruler difference packings

Fix an integer $t\geq 3$ . For a $t$ -element set $A\subset\mathbb{Z}$ , let $\Delta(A)=\{a_j-a_i:a_i,a_j\in A,\ a_i<a_j\}$ be its set of positive differences. Call $A$ a Golomb ruler if all elements of $\Delta(A)$ are distinct. For a family $\mathcal{F}$ of $t$ -mark Golomb rulers whose difference sets are pairwise disjoint and satisfy $\bigcup_{A\in\mathcal{F}}\Delta(A)\subseteq[1,U]\cap\mathbb{Z}$ , define $P_t(U)$ to be the maximum possible value of $\left|\bigcup_{A\in\mathcal{F}}\Delta(A)\right|$ . The claimed threshold theorem is $P_t(U)=U-o(U)$ as $U\to\infty$ if and only if $3\leq t\leq 5$ . For every fixed $t\geq 6$ , the claimed obstruction is $\liminf_{U\to\infty}\left(1-\frac{P_t(U)}{U}\right)\geq\frac{(t-1)\gamma_0-2}{2(t-2)}$ , where $x_0\in(\pi,3\pi/2)$ is the first positive solution of $\tan x=x$ and $\gamma_0=-2\sin(x_0)/x_0=0.4344672564\ldots$ .

References

Primary source

arXiv

Additional references

An Almost-Covering Threshold for Golomb-Ruler Difference Packings — arXiv — Chaohang Ma, Xiangjie Yi

additive-combinatorics design-theory

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A new unrefereed paper claims to settle when collections of rulers can cover almost every distance, proving a sharp cutoff and a definite gap for six marks.

The problem asks when pairwise difference-disjoint Golomb rulers can cover almost all integers up to a large bound. Chaohang Ma and Xiangjie Yi claim the exact threshold: asymptotically complete coverage occurs precisely for $3\le t\le 5$ .

Known results

The preprint attributes the $t=3,4$ cases to known perfect-difference-family existence results and the $t=5$ case to Wild's product construction, recorded by Mathon, using perfect families at orders $121$ and $161$ .

August 2026 threshold claim

For every fixed $t\ge 6$ , Ma and Yi claim a positive-density uncovered set, with

\liminf_{U\to\infty}\left(1-\frac{P_t(U)}{U}\right)\ge\frac{(t-1)\gamma_0-2}{2(t-2)}.

In particular, they claim an asymptotic uncovered proportion of at least $0.021542035264\ldots$ for $t=6$ . The same paper gives a stronger large- $t$ obstruction, but explicitly remains unrefereed.

Current status (as of August 2026): The threshold theorem and the six-mark quantitative gap are claimed in an unrefereed preprint, but no independent verification is recorded; the mathematical claim therefore remains unconfirmed.

Sources

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