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Proceedings of the American Mathematical Society

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



Certain averages on the $\textbf {a}$-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
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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 \[ A_N^{\left ( 1 \right )}f\left ( {\mathbf {x}} \right ) = \frac {1}{N}\sum \limits _{n = 1}^N {f\left ( {{\mathbf {x}} + {n^2}\alpha } \right ),{\text { and }}A_N^{\left ( 2 \right )}f\left ( {\mathbf {x}} \right ) = \frac {1}{N}\sum \limits _{n = 1}^N {f\left ( {{\mathbf {x}} + {p_n}\alpha } \right ),} } \] 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 $f$.

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  • J. Bourgain, On the maximal ergodic theorem for certain subsets of the integers, Israel J. Math. 61 (1988), no. 1, 39–72. MR 937581, DOI
  • Jean Bourgain, Pointwise ergodic theorems for arithmetic sets, Inst. Hautes Études Sci. Publ. Math. 69 (1989), 5–45. With an appendix by the author, Harry Furstenberg, Yitzhak Katznelson and Donald S. Ornstein. MR 1019960
  • Harold Davenport, Multiplicative number theory, 2nd ed., Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York-Berlin, 1980. Revised by Hugh L. Montgomery. MR 606931
  • Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. I, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 115, Springer-Verlag, Berlin-New York, 1979. Structure of topological groups, integration theory, group representations. MR 551496
  • ---, Abstract harmonic analysis, vol. 2, Springer-Verlag, Berlin, 1970. R. Nair, On polynomials in primes and J. Bourgain’s circle method approach to ergodic theorems, Ergodic Theory Dynamical Systems (to appear). ---, On some arithmetic properties of ${L^p}$ summable functions, preprint.
  • MátĂ© Wierdl, Pointwise ergodic theorem along the prime numbers, Israel J. Math. 64 (1988), no. 3, 315–336 (1989). MR 995574, DOI

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Article copyright: © Copyright 1992 American Mathematical Society