α-expansions, linear recurrences, and the sum-of-digits function

Abstract The higher-dimensional generalization of the weighted q-adic sum-of-digits functions sq,γ (n), n =0, 1, 2,..., covers several important cases of sequences investigated in the theory of uniformly distributed sequences, e.g., d-dimensional van der Corput-Halton or d-dimensional Kronecker sequences. We prove a necessary and sufficient condition for the higher-dimensional weighted q-adic sum-of-digits functions to be uniformly distributed modulo one in terms of a trigonometric product. As applications of our condition we prove some upper estimates of the extreme discrepancies of such sequences, and that the existence of distribution function g(x)= x implies the uniform distribution modulo one of the weighted q-adic sum-of-digits function sq,γ (n), n = 0, 1, 2,... We also prove the uniform distribution modulo one of related sequences h 1 sq, γ (n)+h 2 sq,γ (n +1), where h 1 and h 2 are integers such that h 1 + h 2 ≠ 0 and that the akin two-dimensional sequence sq,γ (n), sq,γ (n +1) cannot be uniformly distributed modulo one if q ≥ 3. The properties of the two-dimensional sequence sq,γ (n),s q,γ (n +1), n =0, 1, 2,..., will be instrumental in the proofs of the final section, where we show how the growth properties of the sequence of weights influence the distribution of values of the weighted sum-of-digits function which in turn imply a new property of the van der Corput sequence.

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Output Sum of Transducers: Limiting Distribution and Periodic Fluctuation

The Electronic Journal of Combinatorics ◽

10.37236/5026 ◽

2015 ◽

Vol 22 (2) ◽

Cited By ~ 1

Author(s):

Clemens Heuberger ◽

Sara Kropf ◽

Helmut Prodinger

Keyword(s):

Periodic Function ◽

Error Term ◽

Fourier Coefficients ◽

Higher Dimensions ◽

Expected Value ◽

Hölder Continuous ◽

Periodic Fluctuation ◽

Sum Of Digits ◽

Sum Of Digits Function ◽

Random Integer

As a generalization of the sum of digits function and other digital sequences, sequences defined as the sum of the output of a transducer are asymptotically analyzed. The input of the transducer is a random integer in $[0, N)$. Analogues in higher dimensions are also considered. Sequences defined by a certain class of recursions can be written in this framework.Depending on properties of the transducer, the main term, the periodic fluctuation and an error term of the expected value and the variance of this sequence are established. The periodic fluctuation of the expected value is Hölder continuous and, in many cases, nowhere differentiable. A general formula for the Fourier coefficients of this periodic function is derived. Furthermore, it turns out that the sequence is asymptotically normally distributed for many transducers. As an example, the abelian complexity function of the paperfolding sequence is analyzed. This sequence has recently been studied by Madill and Rampersad.

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Statistical independence in mathematics–the key to a Gaussian law

Mathematische Semesterberichte ◽

10.1007/s00591-020-00287-z ◽

2020 ◽

Author(s):

Gunther Leobacher ◽

Joscha Prochno

Keyword(s):

Central Limit Theorem ◽

Limit Theorem ◽

Central Limit ◽

Historical Context ◽

Lacunary Series ◽

Statistical Independence ◽

Binary Expansion ◽

Sum Of Digits ◽

Sum Of Digits Function ◽

Kolmogorov Axioms

Abstract In this manuscript we discuss the notion of (statistical) independence embedded in its historical context. We focus in particular on its appearance and role in number theory, concomitantly exploring the intimate connection of independence and the famous Gaussian law of errors. As we shall see, this at times requires us to go adrift from the celebrated Kolmogorov axioms, which give the appearance of being ultimate ever since they have been introduced in the 1930s. While these insights are known to many a mathematician, we feel it is time for both a reminder and renewed awareness. Among other things, we present the independence of the coefficients in a binary expansion together with a central limit theorem for the sum-of-digits function as well as the independence of divisibility by primes and the resulting, famous central limit theorem of Paul Erdős and Mark Kac on the number of different prime factors of a number $$n\in{\mathbb{N}}$$ n ∈ N . We shall also present some of the (modern) developments in the framework of lacunary series that have its origin in a work of Raphaël Salem and Antoni Zygmund.

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A lower bound for a remainder term associated with the sum of digits function

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500005058 ◽

1991 ◽

Vol 34 (1) ◽

pp. 121-142 ◽

Cited By ~ 2

Author(s):

D. M. E. Foster

Keyword(s):

Lower Bound ◽

Positive Integer ◽

Upper Bound ◽

Remainder Term ◽

Fixed Integer ◽

Sum Of Digits ◽

Sum Of Digits Function ◽

The Difference

For a fixed integer q≧2, every positive integer k = Σr≧0ar(q, k)qr where each ar(q, k)∈{0,1,2,…, q−1}. The sum of digits function α(q, k) Σr≧0ar(q, k) behaves rather erratically but on averaging has a uniform behaviour. In particular if , where n>1, then it is well known that A(q, n)∼½((q − 1)/log q)n logn as n → ∞. For odd values of q, a lower bound is now obtained for the difference 2S(q, n) = A(q, n)−½(q − 1))[log n/log q, where [log n/log q] denotes the greatest integer ≦log n /log q. This complements an upper bound already found.

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