Lowness for effective Hausdorff dimension

Abstract

Abstract. We examine the sequences A that are low for dimension, i.e., those for which the effective (Hausdorff) dimension relative to A is the same as the unrelativized effective dimension. Lowness for dimension is a weakening of lowness for randomness, a central notion in effective randomness. By considering analogues of characterizations of lowness for randomness, we show that lowness for dimension can be characterized in several ways. It is equivalent to lowishness for randomness, namely, that every Martin-Löf random sequence has effective dimension 1 relative to A, and lowishness for K, namely, that the limit ofWe show that there is a perfect Π 0 1 -class of low for dimension sequences. Since there are only countably many low for random sequences, many more sequences are low for dimension. Finally, we prove that every low for dimension is jump-traceable in order n ε , for any ε > 0.

A Perfect Set of Reals with Finite Self-Information

“…Note that the fact that there is a perfect set of such reals was discovered independently by Lempp, Miller, Ng, Turetsky, and Weber and will appear in their forthcoming paper [4].…”

“…The limit of the log n n term is 0 as n → ∞, so dim(S) = dim A (S) and Dim(S) = Dim A (S). ⊣ Note that the fact that there is a perfect set of such reals was discovered independently by Lempp, Miller, Ng, Turetsky, and Weber and will appear in their forthcoming paper [4]. §6.…”

“…Note that the fact that there is a perfect set of such reals was discovered independently by Lempp, Miller, Ng, Turetsky, and Weber and will appear in their forthcoming paper [4]. 6 Extending to Infinitely Many f 's Theorem 1.7 allows us to build a perfect set of reals that all satisfy K(σ) ≤ + K A (σ) + f (σ) for f 's that have well-behaved recursive approximations.…”

Finite Self-Information

“…The function f from Lemma 2.3 grows much more slowly than log |σ|, so it seems to be much easier to produce a set that is low for dimension than one that has finite self-information. On the other hand, Lempp, Miller, Ng, Turetsky, and Weber [11] show that if a set is low for effective Hausdorff dimension then it is jump traceable via O(h r ) for any convergent order function h and any r > 1, and thus, in particular, via n 1+ for any > 0, which is a better bound than what we obtained above for sets that have finite self-information.…”

“…In the forthcoming paper [11], Lempp, Miller, Ng, Turetsky, and Weber show that there is in fact a perfect Π 0 1 class of sets that are low for effective Hausdorff dimension, a result that also follows by the methods of Herbert [8].…”