The Dimension Spectrum Conjecture for Planar Lines

Abstract

Let $L_{a,b}$ be a line in the Euclidean plane with slope $a$ and intercept $b$. The dimension spectrum $\spec(L_{a,b})$ 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 $d=\min \{\dim (a,b),1\}$. For every $s\in (0,1)$, we construct a point $x$ such that $\dim (x,ax+b)=d+s$. Thus, we show that $\spec(L_{a,b})$ contains the interval $(d,1+d)$. Results of Turetsky , and Lutz and Stull, show that $\spec(L_{a,b})$ contain the endpoints $d$ and $1+d$. Taken together, $[d, 1 + d] \subseteq \spec(L_{a,b})$, for every planar line $L_{a,b}$.