Decidability of the “almost all” theory of degrees

1972 ◽  
Vol 37 (3) ◽  
pp. 501-506 ◽  
Author(s):  
John Stillwell
Keyword(s):  
Lebesgue Measure ◽  
Measure Theory ◽  
Random Sequence ◽  
Recursion Theory ◽  
Sharp Contrast ◽  
Standard Theory ◽  
Simple Manner ◽  
Fubini’S Theorem ◽  
Almost All ◽  
Random Pair

Ever since Spector's brilliant application of measure theory to recursion theory in 1958 [6] it has been realized that measure theory promotes sweeping simplifications in the theory of degrees. Results previously thought to be pathological were shown by Spector, and later Sacks [4], [5], to hold for almost all degrees (“almost all” in the sense of Lebesgue measure), often with much simpler proofs. Good examples of this phenomenon are Spector's demonstration that almost all pairs of sets are of incomparable degree (as an immediate consequence of Fubini's theorem) and Sacks' exquisitely simple deduction from this result that almost every degree is the join of two incomparable degrees (for if a random sequence is decomposed into its even and odd parts, the result is a random pair).The present paper attempts to vindicate the feeling that almost all degrees behave in a simple manner by showing that if the quantifier in the theory of degrees with ′(jump), ∪ (join) and ∩ (meet) is taken to be (almost ∀a) instead of (∀a) then the theory is decidable. We are able to use ∩ because it will be shown that if t1, t2 are any terms built from degree variables a1, …, am with ′ and ∪ then t1 ∩ t2 exists for almost all a1, …, am. Thus the “almost all” theory presents a sharp contrast to the standard theory, where ∩ is not always defined (Kleene-Post [1]) and which is known to be undecidable (Lachlan [2]).

The determination of a function from its projections

1979 ◽  
Vol 85 (2) ◽  
pp. 351-355 ◽  
Author(s):  
K. J. Falconer
Keyword(s):  
Lebesgue Measure ◽  
Unit Vector ◽  
Fubini’S Theorem ◽  
Perpendicular Distance ◽  
Almost All ◽  
Dimensional Lebesgue Measure ◽  
Fubini's Theorem

Let H(t, θ) be the hyperplane in Rn (n ≥ 2) which is perpendicular to the unit vector θ and perpendicular distance t from the origin, that is H(t, θ) = {x ε Rn: x.θ = t} (Note that H(t, θ) and H(−t, −θ) are the same hyperplane.) If f(x)ε L1(Rn) we will denote by F(t, θ) the projection of f perpendicular to θ, that is the integral of f(x) over H(t, θ) with respect to (n − 1)-dimensional Lebesgue measure. By Fubini's Theorem, if f(x) ε L1 (Rn), F(t, θ) exists for almost all t for every θ.

Consistency conditions for a finite set of projections of a function

1979 ◽  
Vol 85 (1) ◽  
pp. 61-68 ◽  
Author(s):  
K. J. Falconer
Keyword(s):  
Lebesgue Measure ◽  
Convex Subset ◽  
Compact Convex ◽  
Unit Vector ◽  
Compact Convex Subset ◽  
Finite Collection ◽  
Finite Set ◽  
Almost All ◽  
Dimensional Lebesgue Measure ◽  
Unit Vectors

Let H(μ, θ) be the hyperplane in Rn (n ≥ 2) that is perpendicular to the unit vector 6 and perpendicular distance μ from the origin; that is, H(μ, θ) = (x ∈ Rn: x. θ = μ). (Note that H(μ, θ) and H(−μ, −θ) are the same hyperplanes.) Let X be a proper compact convex subset of Rm. If f(x) ∈ L1(X) we will denote by F(μ, θ) the projection of f perpendicular to θ; that is, the integral of f(x) over H(μ, θ) with respect to (n − 1)-dimensional Lebesgue measure. By Fubini's Theorem, if f(x) ∈ L1(X), F(μ, θ) exists for almost all μ for every θ. Our aim in this paper is, given a finite collection of unit vectors θ1, …, θN, to characterize the F(μ, θi) that are the projections of some function f(x) with support in X for 1 ≤ i ≤ N.

The trajectory of the turning point is dense for almost all tent maps

1996 ◽  
Vol 16 (6) ◽  
pp. 1173-1183 ◽  
Author(s):  
Karen Brucks ◽  
Michal Misiurewicz
Keyword(s):  
Lebesgue Measure ◽  
Turning Point ◽  
Tent Map ◽  
Tent Maps ◽  
Almost All

AbstractWe prove that for almost every (with respect to the Lebesgue measure) a ∈ [√2, 2], the forward trajectory of the turning point of the tent map fa with slope a is dense in the interval of transitivity of fa.

A Family of Generalized Riesz Products

1996 ◽  
Vol 48 (2) ◽  
pp. 302-315 ◽  
Author(s):  
A. H. Dooley ◽  
S. J. Eigen
Keyword(s):  
Lebesgue Measure ◽  
Spectral Measure ◽  
Probability Space ◽  
Rank One ◽  
Almost All ◽  
Riesz Products

AbstractGeneralized Riesz products similar to the type which arise as the spectral measure for a rank-one transformation are studied. A condition for the mutual singularity of two such measures is given. As an application, a probability space of transformations is presented in which almost all transformations are singular with respect to Lebesgue measure.

scholarly journals Some Remarks on Boundary Behavior of Analytic and Meromorphic Functions

1955 ◽  
Vol 9 ◽  
pp. 79-85 ◽  
Author(s):  
F. Bagemihl ◽  
W. Seidel
Keyword(s):  
Meromorphic Function ◽  
Uniqueness Theorem ◽  
Meromorphic Functions ◽  
Boundary Behavior ◽  
Regular Function ◽  
Simple Manner ◽  
Regular Functions ◽  
Almost All ◽  
Jordan Arcs

This paper is concerned with regular and meromorphic functions in |z| < 1 and their behavior near |z| = 1. Among the results obtained are the following. In section 2 we prove the existence of a non-constant meromorphic function that tends to zero at every point of |z| = 1 along almost all chords of |z| < 1 terminating in that point. Section 3 deals with the impossibility of ex tending this result to regular functions. In section 4 it is shown that a regular function can tend to infinity along every member of a set of spirals approach ing |z| = 1 and exhausting |z| < 1 in a simple manner. Finally, in section 5 we prove that this set of spirals cannot be replaced by an exhaustive set of Jordan arcs terminating in points of |z| = 1; Theorem 3 of this section can be interpreted as a uniqueness theorem for meromorphic functions.

Some theorems in the metric theory of diophantine approximation

1986 ◽  
Vol 99 (3) ◽  
pp. 385-394 ◽  
Author(s):  
Glyn Harman
Keyword(s):  
Lebesgue Measure ◽  
Diophantine Approximation ◽  
Additional Constraint ◽  
Number Of Solutions ◽  
Initial Question ◽  
Image Position ◽  
Almost All

An excellent introduction to the metric theory of diophantine approximation is provided by [19], where, in chapter 1·7, the reader may find a discussion of the first two problems considered in this paper. Our initial question concerns the number of solutions of the inequalityfor almost all α(in the sense of Lebesgue measure on ℝ). Here ∥ ∥ denotes distance to a nearest integer, {βr}, {ar} are given sequences of reals and distinct integers respectively, and f is a function taking values in [0, ½] and with Σf(r) divergent (for convenience we write ℱ for the set of all such functions). It is reasonable to expect that, for almost all α and with some additional constraint on f, the number of solutions of (1) is asymptotically equal toas k tends to infinity.

scholarly journals On Perimeters and Volumes of Fattened Sets

2019 ◽  
Vol 2019 ◽  
pp. 1-12
Author(s):  
Andrea C. G. Mennucci
Keyword(s):  
Lebesgue Measure ◽  
Measure Theory ◽  
Geometric Measure Theory ◽  
Coarea Formula ◽  
Compact Set ◽  
Geometric Measure ◽  
General Bound

In this paper we analyze the shape of fattened sets; given a compact set C⊂RN let Cr be its r-fattened set; we prove a general bound rP(Cr)≤NL({Cr∖C}) between the perimeter of Cr and the Lebesgue measure of Cr∖C. We provide two proofs: one elementary and one based on Geometric Measure Theory. Note that, by the Flemin–Rishel coarea formula, P(Cr) is integrable for r∈(0,a). We further show that for any integrable continuous decreasing function ψ:(0,1/2)→(0,∞) there exists a compact set C⊂RN such that P(Cr)≥ψ(r). These results solve a conjecture left open in (Mennucci and Duci, 2015) and provide new insight in applications where the fattened set plays an important role.

Measures: Back and Forth Between Point sets and Large sets

1995 ◽  
Vol 1 (2) ◽  
pp. 170-188 ◽  
Author(s):  
Noa Goldring
Keyword(s):  
Lebesgue Measure ◽  
Set Theory ◽  
Real Line ◽  
Measure Theory ◽  
Impossibility Result ◽  
Point Sets ◽  
Large Sets ◽  
Starting Point ◽  
Standard Set ◽  
Action At A Distance

It was questions about points on the real line that initiated the study of set theory. Points paved the way to point sets and these to ever more abstract sets. And there was more: Reflection on structural properties of point sets not only initiated the study of ordinary sets; it also supplied blueprints for defining extra-ordinary, “large” sets, transcending those provided by standard set theory. In return, the existence of such large sets turned out critical to settling open conjectures about point sets.How to explain such action at a distance between the very large and the rather small? Rather than having an air of magic, could these results rest on deep structural similarities between the two superficially distant species of sets?In this essay I dissect one group of such two-way results. Their linchpin is the notion of measure.§1. Vitali's impossibility result. Our starting point is a problem in measure theory regarding the notion of “Lebesgue measure.” Before presenting the problem, I would like to review the notion of Lebesgue measure. Rather than listing its main properties, I would like to show how Lebesgue measure is born out of an attempt to generalize the notion of the length of an interval to arbitrary sets of reals. One tries to approximate arbitrary sets of reals by intervals, in the hope that the lengths of the intervals will induce a measure on these sets.

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