Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Certain averages on the $ {\bf a}$-adic numbers

Authors: Nakhlé H. Asmar and Radhakrishnan Nair
Journal: Proc. Amer. Math. Soc. 114 (1992), 21-28
MSC: Primary 11K41; Secondary 11K55, 22D40, 28D99, 47A35
MathSciNet review: 1087460
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For $ {L^p} \cap {L^2}$ functions $ f$, with $ p$ greater than one, defined on the $ {\mathbf{a}}$-adic numbers $ {\Omega _{\mathbf{a}}}$, we consider averages like

$\displaystyle A_N^{\left( 1 \right)}f\left( {\mathbf{x}} \right) = \frac{1}{N}\... ...{1}{N}\sum\limits_{n = 1}^N {f\left( {{\mathbf{x}} + {p_n}\alpha } \right),} } $

where $ {\mathbf{x}}$ and $ \alpha $ are in $ {\Omega _{\mathbf{a}}}$. Here $ {p_n}$ denotes the $ n$th prime. These averages are known to converge for almost all $ {\mathbf{x}}$. We describe explicitly these limits, which possibly contrary to expectation, turn out in general not to be the integral of $ f$.

References [Enhancements On Off] (What's this?)

  • [1] J. Bourgain, On the maximal ergodic theorem for certain subsets of the integers, Israel J. Math. 61 (1988), 38-72. MR 937581 (89f:28037a)
  • [2] -, Pointwise ergodic theorems for arithmetic sets, Inst. Hautes Études Sci. Publ. Math. 69 (1989), 5-45. MR 1019960 (90k:28030)
  • [3] H. Davenport, Multiplicative number theory, 2nd ed. Graduate Texts in Math., vol. 74, Springer-Verlag, 1980. MR 606931 (82m:10001)
  • [4] E. Hewitt and K. A. Ross, Abstract harmonic analysis, vol. 1, 2nd ed., Springer-Verlag, Berlin, 1979. MR 551496 (81k:43001)
  • [5] -, Abstract harmonic analysis, vol. 2, Springer-Verlag, Berlin, 1970.
  • [6] R. Nair, On polynomials in primes and J. Bourgain's circle method approach to ergodic theorems, Ergodic Theory Dynamical Systems (to appear).
  • [7] -, On some arithmetic properties of $ {L^p}$ summable functions, preprint.
  • [8] M. Wierdl, Pointwise ergodic theorems along the prime numbers, Israel J. Math. 64 (1988), 315-336. MR 995574 (90f:11062)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 11K41, 11K55, 22D40, 28D99, 47A35

Retrieve articles in all journals with MSC: 11K41, 11K55, 22D40, 28D99, 47A35

Additional Information

Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society