Abstract
This paper investigates the algorithmic dimension spectra of lines in the Euclidean plane. Given any line L with slope a and vertical intercept b, the dimension spectrum \({{\mathrm{sp}}}(L)\) is the set of all effective Hausdorff dimensions of individual points on L. We draw on Kolmogorov complexity and geometrical arguments to show that if the effective Hausdorff dimension \(\dim (a, b)\) is equal to the effective packing dimension \({{\mathrm{Dim}}}(a, b)\), then \({{\mathrm{sp}}}(L)\) contains a unit interval. We also show that, if the dimension \(\dim (a, b)\) is at least one, then \({{\mathrm{sp}}}(L)\) is infinite. Together with previous work, this implies that the dimension spectrum of any line is infinite.
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