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A Golomb ruler can be described as a set of marks on a line having integer positions, such that no two pairs of marks are separated by the same distance. Call the spectrum of a ruler the (multi-)set of distances measured by the ruler. For example, consider a ruler $G_3$ on $n=3+1$ marks, with marks at $\{0,1,4,6\}$. The spectrum of $G_3$ is $\{1,2,3,4,5,6\}$. In this case, $G_3$ is perfect, in that every distance between $1$ and $6$ inclusive can be measured.

Define the $L^2$ norm of a set of marks as the square root of the sum of squares of the multiplicity of the spectrum. For example, a ruler with marks at $\{0,1,3,5,8\}$ has a spectrum of $\{1,2,2,3,3,4,5,5,7,8\}$, and thus an $L^2$ norm of $(1^2+2^2+2^2+1^2+2^2+1^2+1^2)^{1/2}=4$.

The requirement of having no two distances measured twice seems overly narrowing. For example, there may be some applications, in, say, radio astronomy, where allowing a repeat measurement every now and then may be acceptable. Thus, I consider a "generalized" optimal ruler $G_{m,k}$ (not great name) that, instead of having no two pairs of marks separated by the same distance, minimizes such an $L^2$ norm over all rulers on $m$ marks of length $k$. For a Golomb ruler, the $L^2$ norm of an $(m+1)$-mark ruler is $\sqrt{m(m-1)/2}$, but there might be clever constructions to stuff marks onto a ruler that has a nice spectrum with small $l^2$ norm.

Have such rulers been studied, perhaps in relation to Sidon sets? Is there anything nontrivial known about the relation of the $L^2$ norm to the number of marks $m$ or the length of the ruler $k$?

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  • $\begingroup$ Have you computed the norm for extreme examples, like say a ruler from the hardware store? It may be the L2 norm is not fine enough. Gerhard "Perhaps Measure A Different Way?" Paseman, 2018.11.20. $\endgroup$ – Gerhard Paseman Nov 20 '18 at 23:59

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