Search results for "Pigeonhole principle"
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If a DOL language is k-power free then it is circular
1993
We prove that if a DOL language is k-power free then it is circular. By using this result we are able to give an algorithm which decides whether, fixed an integer k≥1, a DOL language is k-power free; we are also able to give a new simpler proof of a result, previously obtained by Ehrenfeucht and Rozenberg, that states that it is decidable whether a DOL language is k-power free for some integer k≥1.
Combinatorial proofs of two theorems of Lutz and Stull
2021
Recently, Lutz and Stull used methods from algorithmic information theory to prove two new Marstrand-type projection theorems, concerning subsets of Euclidean space which are not assumed to be Borel, or even analytic. One of the theorems states that if $K \subset \mathbb{R}^{n}$ is any set with equal Hausdorff and packing dimensions, then $$ \dim_{\mathrm{H}} π_{e}(K) = \min\{\dim_{\mathrm{H}} K,1\} $$ for almost every $e \in S^{n - 1}$. Here $π_{e}$ stands for orthogonal projection to $\mathrm{span}(e)$. The primary purpose of this paper is to present proofs for Lutz and Stull's projection theorems which do not refer to information theoretic concepts. Instead, they will rely on combinatori…