Sparse Ruler #
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A sparse ruler of length $L$ is a sequence of marks $0 = a_1 < a_2 < \dots < a_m = L$. A distance $k \in \mathbb{N}$ can be measured if there are $i, j \in \{1, \dots, m\}$, such that $k = a_j - a_i$.
One can now ask for rulers that measure every integer up to some $K \in \mathbb{N}$ and for them to be minimal, i.e. having a minimal number of marks. Furthermore, we can restrict such rulers in length, for example requiring for a ruler of length $L$ to measure every distance up to $L$. This is called a perfect ruler and Erdős Problem 170 covers the question of how many marks such minimum perfect rulers have asymptotically.
There are several other questions with regards to sparse rulers and many of them are still unsolved.