LOWNESS FOR EFFECTIVE HAUSDORFF DIMENSION 1 ...
LOWNESS FOR EFFECTIVE HAUSDORFF DIMENSION STEFFEN LEMPP, JOSEPH S. MILLER, KENG MENG NG, DANIEL D. TURETSKY, AND REBECCA WEBER
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-L¨ of random sequence has effective dimension 1 relative to A, and lowishness for K, namely, that the limit of K A (n)/K(n) is 1. We show that there is a perfect Π01 -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 nε , for any ε > 0.
1. Introduction The effective (Hausdorff ) dimension of an infinite sequence A ∈ 2ω is K(A n) . n This notion was introduced by Lutz [15], who effectivized a martingale characterization of Hausdorff dimension and defined dim(A) to be the effective Hausdorff dimension of {A}. The characterization above, which reveals effective dimension to be a natural measure of the information density of infinite sequences, was given by Mayordomo [16] (see also Ryabko [20]). We say that A ∈ 2ω is low for (effective Hausdorff ) dimension if dim(A) = lim inf n→∞
(∀X ∈ 2ω )[dimA (X) ≥ dim(X)]. In other words, A is too weak as an oracle to change the effective Hausdorff dimension of any sequence. It is clear that dimA (X) ≤ dim(X), for any A ∈ 2ω , so if A is low for dimension we have dimA (X) = dim(X) for all X. This paper initiates the study of lowness for dimension. Our main result gives several characterizations of the notion. These can be seen as weakenings of wellknown characterizations of lowness for randomness. This analogy helped direct our research and will be discussed in detail below. Date: December 11, 2014. 2010 Mathematics Subject Classification. Primary 03D32; Secondary 68Q30, 28A78. Lempp’s research was partly supported by Grant # 13407 by the John Templeton Foundation entitled “Exploring the Infinite by Finitary Means” as well as AMS-Simons Foundation Collaboration Grant 209087. Miller was supported by the National Science Foundation under grants DMS-0945187. Miller was also supported by NSF grant DMS-0946325 and Weber by NSF grant DMS-0652326, both part of a Focused Research Group in Algorithmic Randomness. 1
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LEMPP, MILLER, NG, TURETSKY, AND WEBER
We need a few more definitions before we can state the main theorem. If σ ∈ 2
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-L¨ of random sequence has effective dimension 1 relative to A, and lowishness for K, namely, that the limit of K A (n)/K(n) is 1. We show that there is a perfect Π01 -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 nε , for any ε > 0.
1. Introduction The effective (Hausdorff ) dimension of an infinite sequence A ∈ 2ω is K(A n) . n This notion was introduced by Lutz [15], who effectivized a martingale characterization of Hausdorff dimension and defined dim(A) to be the effective Hausdorff dimension of {A}. The characterization above, which reveals effective dimension to be a natural measure of the information density of infinite sequences, was given by Mayordomo [16] (see also Ryabko [20]). We say that A ∈ 2ω is low for (effective Hausdorff ) dimension if dim(A) = lim inf n→∞
(∀X ∈ 2ω )[dimA (X) ≥ dim(X)]. In other words, A is too weak as an oracle to change the effective Hausdorff dimension of any sequence. It is clear that dimA (X) ≤ dim(X), for any A ∈ 2ω , so if A is low for dimension we have dimA (X) = dim(X) for all X. This paper initiates the study of lowness for dimension. Our main result gives several characterizations of the notion. These can be seen as weakenings of wellknown characterizations of lowness for randomness. This analogy helped direct our research and will be discussed in detail below. Date: December 11, 2014. 2010 Mathematics Subject Classification. Primary 03D32; Secondary 68Q30, 28A78. Lempp’s research was partly supported by Grant # 13407 by the John Templeton Foundation entitled “Exploring the Infinite by Finitary Means” as well as AMS-Simons Foundation Collaboration Grant 209087. Miller was supported by the National Science Foundation under grants DMS-0945187. Miller was also supported by NSF grant DMS-0946325 and Weber by NSF grant DMS-0652326, both part of a Focused Research Group in Algorithmic Randomness. 1
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LEMPP, MILLER, NG, TURETSKY, AND WEBER
We need a few more definitions before we can state the main theorem. If σ ∈ 2
