# Has anyone studied Golomb rulers having a spectrum with a minimal $L^2$ norm?

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 $$k$$?

• 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. – Gerhard Paseman Nov 20 '18 at 23:59