The Dimension Spectrum Conjecture for Planar Lines

01/30/2021

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by   D. M. Stull, et al.

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Let L_a,b be a line in the Euclidean plane with slope a and intercept b. The dimension spectrum (L_a,b) is the set of all effective dimensions of individual points on L_a,b. The dimension spectrum conjecture states that, for every line L_a,b, the spectrum of L_a,b contains a unit interval. In this paper we prove that the dimension spectrum conjecture is true. Let (a,b) be a slope-intercept pair, and let d = min{(a,b), 1}. For every s ∈ (0, 1), we construct a point x such that (x, ax + b) = d + s. Thus, we show that (L_a,b) contains the interval (d, 1+ d). Results of Turetsky , and Lutz and Stull, show that (L_a,b) contain the endpoints d and 1+d. Taken together, [d, 1 + d] ⊆(L_a,b), for every planar line L_a,b.