Erdős Problem #170
sparse ruler problem
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
An \(N\)-perfect ruler is a finite subset \(A \subseteq \mathbb{N}\) (the marks), such that each positive integer \(k \leq N\) can be measured, that is, expressed as a difference \(k = a_1 - a_0\) with \(a_0, a_1 \in A\). The set \(A\) is then also called a difference basis w.r.t. \(N\).
We define the set of all \(N\)-perfect rulers \(A\) of length \(N\), i.e. subsets \(A \subseteq \{0, \dots, N\}\), s.t. \(A\) is \(N\)-perfect. This is also called a restricted difference basis w.r.t. \(N\).
The trivial ruler with all marks \(\{0, \dots, N\}\).
Attempts
submit one →If you take a run at this one — with your own AI subscription or without — the guide says how, and the submission form gets you an honest read of what you found.