The sparse ruler constant

Find the limit of F(N)/√N, where F(N) is the fewest marks on a ruler of length N that measures every distance; it lies in [1.56, √3].

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1	# The sparse ruler constant
2
3	A ruler of length $N$ with marks at $0$ and $N$ can measure every whole-number distance up to $N$ using only about $\sqrt{3N}$ marks if they are placed cleverly. How few marks are really needed? Erdős and Gál showed that the minimum, divided by $\sqrt N$, tends to a limit, but its value is unknown: it lies between $1.56$ and $\sqrt3\approx1.732$.
4
5	## Problem
6
7	Let $F(N)$ be the smallest size of a set $A\subseteq\{0,1,\dots,N\}$ such that every integer $0\le d\le N$ is a difference of two elements of $A$, that is, $\{0,1,\dots,N\}\subseteq A-A$. Such a set is a sparse (or complete) ruler of length $N$. Determine
8
9	$$
10	\lim_{N\to\infty}\frac{F(N)}{\sqrt N}.
11	$$
12
13	## Background
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15	Rédei asked whether this limit exists, and Erdős and Gál proved in 1948 that it does. Leech (1956) improved the bounds and showed that the limit equals $\inf_N (F(N)+2)/\sqrt{N+1}$, so every explicit ruler yields a rigorous upper bound. Wichmann (1963) found an infinite family of rulers that gives the upper bound $\sqrt3$. A similar question can be asked for difference bases that are not required to lie inside $[0,N]$.
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17	## What is known
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19	- Bounds. The limit lies in $[1.56,\sqrt3]$: the lower bound is due to Leech (1956), the upper bound to Wichmann's construction (1963).
20	- Computation. Computational evidence collected by Pegg (2020) suggests that Wichmann's bound $\sqrt3$ is the truth. By Leech's infimum formula, if the limit equals $\sqrt3$ then $F(N)=\sqrt{3N}+O(1)$.
21	- Unrestricted version. For difference bases that need not lie in $[0,N]$, Bernshteyn and Tait (2019) raised Leech's lower bound by a small amount.
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23	## What would count as progress
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25	- A ruler $A$ of some length $N$ with $(\lvert A\rvert+2)/\sqrt{N+1}<\sqrt3$, which by Leech's formula would beat Wichmann's bound; it is checked by computing $A-A$.
26	- A lower bound above $1.56$ for the restricted problem, with a complete proof or a certified computation.
27	- Exact values of $F(N)$ for new ranges of $N$, with certificates of optimality.
28	- A formal proof against the Lean statement in [formal-conjectures](https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/170.lean).
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30	## References
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32	1. J. Leech, On the representation of $1,2,\dots,n$ by differences, J. London Math. Soc. 31 (1956), 160–169. [doi:10.1112/jlms/s1-31.2.160](https://doi.org/10.1112/jlms/s1-31.2.160)
33	2. B. Wichmann, A note on restricted difference bases, J. London Math. Soc. 38 (1963), 465–466. [doi:10.1112/jlms/s1-38.1.465](https://doi.org/10.1112/jlms/s1-38.1.465)
34	3. A. Bernshteyn and M. Tait, Improved lower bound for difference bases, J. Number Theory 205 (2019), 50–58. [doi:10.1016/j.jnt.2019.05.002](https://doi.org/10.1016/j.jnt.2019.05.002)
35	4. E. Pegg, Hitting all the marks: exploring new bounds for sparse rulers and a Wolfram Language proof, Wolfram Blog (2020). [blog.wolfram.com](https://blog.wolfram.com/2020/02/12/hitting-all-the-marks-exploring-new-bounds-for-sparse-rulers-and-a-wolfram-language-proof/)
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37	## Source
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39	Imported from [Erdős Problems 170](https://www.erdosproblems.com/170), curated by Thomas Bloom. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.