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- W3127789063 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 $L_{a,b}$." @default.
- W3127789063 created "2021-02-15" @default.
- W3127789063 creator A5004582623 @default.
- W3127789063 date "2021-01-29" @default.
- W3127789063 modified "2023-10-17" @default.
- W3127789063 title "The Dimension Spectrum Conjecture for Planar Lines" @default.
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- W3127789063 doi "https://doi.org/10.48550/arxiv.2102.00134" @default.
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