A generalisation of q-additive functions
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Ladislav Mišík
Abstract
A new concept of the 𝔎-additive arithmetical function is introduced, generalizing the notion of the q-additive function. The properties of the 𝔎-additive arithmetical functions and their impact on the uniform distribution of sequences of the weighted sum-of-digits functions of elements of sequences generated by 𝔎-additive arithmetical functions are studied.
Acknowledgement
The authors would like to thank Oto Strauch for the suggestions and inspirations that led to the presented results.
(Communicated by Milan Paštéka)
References
[1] Bellman, R.−Shapiro, H. N.: On a problem in additive number theory, Ann. Math. (2) 49 (1948), 33–340.10.2307/1969281Search in Google Scholar
[2] Cantor, G.: Über die einfachen Zahlensysteme, Z. Math. Phys. 14 (1869), 121–128.Search in Google Scholar
[3] Fraenkel, A. S.: Systems of numeration, Amer. Math. Monthly 92 (1985), 105–114.10.1080/00029890.1985.11971550Search in Google Scholar
[4] Gelfond, A. O.: Sur les nombres qui ont des propriétés additives et multiplicatives données, Acta Arith. 13 (1968), 259–265.10.4064/aa-13-3-259-265Search in Google Scholar
[5] Haddley, A.−Lertchoosakul, P.−Nair, R.: The Halton sequence and its discrepancy in the Cantor expansion, Period. Math. Hungar. 75 (2017), 128–141.10.1007/s10998-016-0169-5Search in Google Scholar
[6] Hofer, R.−Pillichshammer, F.−Pirsic, G.: Distribution properties of sequences generated by Q-additive functions with respect to Cantor representation of integers, Acta Arith. 138 (2009), 179–200.10.4064/aa138-2-6Search in Google Scholar
[7] Mahler, K.: The spectrum of an array and its application to the study of the translation properties of a simple class of arithmetical functions. II: On the translation properties of a simple class of arithmetical functions, J. Math. Phys. 6 (1927), 158–163; reprinted in Publ. MIT Ser. II 62 No. 118 (1927)).Search in Google Scholar
[8] Mendès France, M.: Les suites a spectre vide et la répartition modulo 1, J. Number Theory 5 (1973), 1–15.10.1016/0022-314X(73)90053-XSearch in Google Scholar
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Articles in the same Issue
- Joins of normal matrices, their spectrum, and applications
- On isbell’s density theorem for bitopological pointfree spaces II
- Every positive integer is a sum of at most n + 2 centered n-gonal numbers
- A generalisation of q-additive functions
- Generalised class groups in dihedral and fake ℤp-extensions
- Singular value bounds with applications to norm and numerical radius inequalities
- Coefficient bounds for convex functions associated with cosine function
- On solutions to q-difference equations for q-appell functions in the spirit of Olsson and Exton
- Numerical implementation of solving a boundary value problem including both delay and parameter
- Solutions of second order iterative boundary value problems with nonhomogeneous boundary conditions
- Global attractivity of nonlinear delay dynamic equations on time scales via Lyapunov functional method
- On pseudo almost periodic solutions of the parabolic-elliptic Keller-Segel systems
- A note on derivations into annihilators of the ideals of banach algebras
- Characterization of nonlinear mixed skew lie and jordan n-Type derivation on ∗-Algebras
- Linear and uniformly continuous surjections between Cp-spaces over metrizable spaces
- A goodness-of-fit test for testing exponentiality based on normalized dynamic survival extropy
- Monotonicity results of ratio between two normalized remainders of Maclaurin series expansion for square of tangent function
