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

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- by NakhlĂ© H. Asmar and Radhakrishnan Nair PDF
- Proc. Amer. Math. Soc.
**114**(1992), 21-28 Request permission

## 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 $f$.## References

- J. Bourgain,
*On the maximal ergodic theorem for certain subsets of the integers*, Israel J. Math.**61**(1988), no.Â 1, 39â€“72. MR**937581**, DOI 10.1007/BF02776301 - Jean Bourgain,
*Pointwise ergodic theorems for arithmetic sets*, Inst. Hautes Ă‰tudes Sci. Publ. Math.**69**(1989), 5â€“45. With an appendix by the author, Harry Furstenberg, Yitzhak Katznelson and Donald S. Ornstein. MR**1019960**, DOI 10.1007/BF02698838 - 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**, DOI 10.1007/978-1-4757-5927-3 - 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**, DOI 10.1007/978-1-4419-8638-2
â€”, - MĂˇtĂ© Wierdl,
*Pointwise ergodic theorem along the prime numbers*, Israel J. Math.**64**(1988), no.Â 3, 315â€“336 (1989). MR**995574**, DOI 10.1007/BF02882425

*Abstract harmonic analysis*, vol. 2, Springer-Verlag, Berlin, 1970. R. Nair,

*On polynomials in primes and J. Bourgainâ€™s circle method approach to ergodic theorems*, Ergodic Theory Dynamical Systems (to appear). â€”,

*On some arithmetic properties of*${L^p}$

*summable functions*, preprint.

## Additional Information

- © Copyright 1992 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**114**(1992), 21-28 - MSC: Primary 11K41; Secondary 11K55, 22D40, 28D99, 47A35
- DOI: https://doi.org/10.1090/S0002-9939-1992-1087460-6
- MathSciNet review: 1087460