A basis theorem for classes of positive measure and jump inversion for random reals

Authors:
Rod Downey and Joseph S. Miller

Journal:
Proc. Amer. Math. Soc. **134** (2006), 283-288

MSC (2000):
Primary 03D28, 68Q30

Published electronically:
August 11, 2005

MathSciNet review:
2170569

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Abstract | References | Similar Articles | Additional Information

Abstract: We extend the Shoenfield jump inversion theorem to the members of any class with nonzero measure; i.e., for every set , there is a real such that . In particular, we get jump inversion for -random reals.

**1.**M. M. Arslanov, R. F. Nadyrov, and V. D. Solov′ev,*A criterion for the completeness of recursively enumerable sets, and some generalizations of a fixed point theorem*, Izv. Vysš. Učebn. Zaved. Matematika**4 (179)**(1977), 3–7 (Russian). MR**0446930****2.**Douglas Cenzer,*Π⁰₁ classes in computability theory*, Handbook of computability theory, Stud. Logic Found. Math., vol. 140, North-Holland, Amsterdam, 1999, pp. 37–85. MR**1720779**, 10.1016/S0049-237X(99)80018-4**3.**R. Downey and D. Hirschfeldt,*Algorithmic randomness and complexity*, Springer-Verlag, Berlin, to appear.**4.**Rodney G. Downey, Denis R. Hirschfeldt, André Nies, and Sebastiaan A. Terwijn,*Calibrating randomness*, Bulletin of Symbolic Logic, to appear.**5.**Richard Friedberg,*A criterion for completeness of degrees of unsolvability*, J. Symb. Logic**22**(1957), 159–160. MR**0098025****6.**Péter Gács,*Every sequence is reducible to a random one*, Inform. and Control**70**(1986), no. 2-3, 186–192. MR**859105**, 10.1016/S0019-9958(86)80004-3**7.**Carl G. Jockusch Jr. and Robert I. Soare,*Π⁰₁ classes and degrees of theories*, Trans. Amer. Math. Soc.**173**(1972), 33–56. MR**0316227**, 10.1090/S0002-9947-1972-0316227-0**8.**S. Kautz,*Degrees of random sets*, Ph.D. Dissertation, Cornell University, 1991.**9.**Antonín Kučera,*Measure, Π⁰₁-classes and complete extensions of 𝑃𝐴*, Recursion theory week (Oberwolfach, 1984) Lecture Notes in Math., vol. 1141, Springer, Berlin, 1985, pp. 245–259. MR**820784**, 10.1007/BFb0076224**10.**Antonín Kučera,*On the use of diagonally nonrecursive functions*, Logic Colloquium ’87 (Granada, 1987) Stud. Logic Found. Math., vol. 129, North-Holland, Amsterdam, 1989, pp. 219–239. MR**1015310**, 10.1016/S0049-237X(08)70130-7**11.**S. Kurtz,*Randomness and genericity in the degrees of unsolvability*, Ph.D. Dissertation, University of Illinois, Urbana, 1981.**12.**Ming Li and Paul Vitányi,*An introduction to Kolmogorov complexity and its applications*, Texts and Monographs in Computer Science, Springer-Verlag, New York, 1993. MR**1238938****13.**Per Martin-Löf,*The definition of random sequences*, Information and Control**9**(1966), 602–619. MR**0223179****14.**Joseph S. Miller and Liang Yu,*On initial segment complexity and degrees of randomness*, to appear.**15.**Gerald E. Sacks,*A minimal degree less than 0’*, Bull. Amer. Math. Soc.**67**(1961), 416–419. MR**0126380**, 10.1090/S0002-9904-1961-10652-6**16.**Dana Scott,*Algebras of sets binumerable in complete extensions of arithmetic*, Proc. Sympos. Pure Math., Vol. V, American Mathematical Society, Providence, R.I., 1962, pp. 117–121. MR**0141595****17.**J. R. Shoenfield,*On degrees of unsolvability*, Ann. of Math. (2)**69**(1959), 644–653. MR**0105355****18.**Robert I. Soare,*Recursively enumerable sets and degrees*, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1987. A study of computable functions and computably generated sets. MR**882921****19.**Frank Stephan,*Martin-Löf random sets and PA-complete sets*, to appear.

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Additional Information

**Rod Downey**

Affiliation:
School of Mathematical and Computing Sciences, Victoria University, P.O. Box 600, Wellington, New Zealand

Email:
Rod.Downey@mcs.vuw.ac.nz

**Joseph S. Miller**

Affiliation:
Department of Mathematics, University of Connecticut, U-3009, 196 Auditorium Road, Storrs, Connecticut 06269

Email:
joseph.s.miller@gmail.com

DOI:
https://doi.org/10.1090/S0002-9939-05-07901-3

Received by editor(s):
April 13, 2004

Received by editor(s) in revised form:
June 30, 2004

Published electronically:
August 11, 2005

Additional Notes:
Both authors were supported by the Marsden Fund of New Zealand. The first author was also partially supported by NSFC Grand International Joint Project Grant No. 60310213 “New Directions in Theory and Applications of Models of Computation” (China).

Communicated by:
Carl G. Jockusch, Jr.

Article copyright:
© Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.