scholarly journals Averaging the sum of digits function to an even base

1992 ◽  
Vol 35 (3) ◽  
pp. 449-455 ◽  
Author(s):  
D. M. E. Foster
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Fixed Integer ◽  
Sum Of Digits ◽  
Sum Of Digits Function ◽  
The Difference

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

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

1991 ◽  
Vol 34 (1) ◽  
pp. 121-142 ◽  
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.

scholarly journals Estimates for a remainder term associated with the sum of digits function

1987 ◽  
Vol 29 (1) ◽  
pp. 109-129 ◽  
Author(s):  
D. M. E. Foster
Keyword(s):  
Positive Integer ◽  
Remainder Term ◽  
Fixed Integer ◽  
Sum Of Digits ◽  
Sum Of Digits Function ◽  
Image Position ◽  
Average Behaviour

If q(≥2) is a fixed integer it is well known that every positive integer k may be expressed uniquely in the formWe introduce the ‘sum of digits’ functionBoth the above sums are of course finite. Although the behaviour of α(q, k) is somewhat erratic, its average behaviour is more regular and has been widely studied.

scholarly journals On the difference of values of the kernel function at consecutive integers

2003 ◽  
Vol 2003 (67) ◽  
pp. 4249-4262
Author(s):  
Jean-Marie De Koninck ◽  
Florian Luca
Keyword(s):  
Positive Integer ◽  
Kernel Function ◽  
Large Family ◽  
Fixed Integer ◽  
Consecutive Integers ◽  
The Difference

For each positive integern, setγ(n)=Πp|np. Given a fixed integerk≠±1, we establish that if theABC-conjecture holds, then the equationγ(n+1)−γ(n)=khas only finitely many solutions. In the particular casesk=±1, we provide a large family of solutions for each of the corresponding equations.

scholarly journals A lower bound for Cusick’s conjecture on the digits of n + t

2021 ◽  
pp. 1-23
Author(s):  
LUKAS SPIEGELHOFER
Keyword(s):  
Lower Bound ◽  
Probability Distribution ◽  
Nonnegative Integer ◽  
Asymptotic Density ◽  
Binary Expansion ◽  
Sum Of Digits ◽  
Sum Of Digits Function

Abstract Let S be the sum-of-digits function in base 2, which returns the number of 1s in the base-2 expansion of a nonnegative integer. For a nonnegative integer t, define the asymptotic density $${c_t} = \mathop {\lim }\limits_{N \to \infty } {1 \over N}|\{ 0 \le n < N:s(n + t) \ge s(n)\} |.$$ T. W. Cusick conjectured that c t > 1/2. We have the elementary bound 0 < c t < 1; however, no bound of the form 0 < α ≤ c t or c t ≤ β < 1, valid for all t, is known. In this paper, we prove that c t > 1/2 – ε as soon as t contains sufficiently many blocks of 1s in its binary expansion. In the proof, we provide estimates for the moments of an associated probability distribution; this extends the study initiated by Emme and Prikhod’ko (2017) and pursued by Emme and Hubert (2018).

scholarly journals ON -K2 FOR NORMAL SURFACE SINGULARITIES

1998 ◽  
Vol 09 (06) ◽  
pp. 653-668 ◽  
Author(s):  
HAO CHEN ◽  
SHIHOKO ISHII
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Rational Number ◽  
Accumulation Point ◽  
Normal Surface ◽  
Accumulation Points ◽  
Surface Singularities

In this paper we show the lower bound of the set of non-zero -K2 for normal surface singularities establishing that this set has no accumulation points from above. We also prove that every accumulation point from below is a rational number and every positive integer is an accumulation point. Every rational number can be an accumulation point modulo ℤ. We determine all accumulation points in [0, 1]. If we fix the value -K2, then the values of pg, pa, mult, embdim and the numerical indices are bounded, while the numbers of the exceptional curves are not bounded.

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

1991 ◽  
Vol 70 (1) ◽  
pp. 311-324 ◽  
Author(s):  
Peter J. Grabner ◽  
Robert F. Tichy
Keyword(s):  
Linear Recurrences ◽  
Sum Of Digits ◽  
Sum Of Digits Function

Uniform Distribution of the Weighted Sum-of-Digits Functions

2021 ◽  
Vol 16 (1) ◽  
pp. 93-126
Author(s):  
Ladislav Mišík ◽  
Štefan Porubský ◽  
Oto Strauch
Keyword(s):  
Uniform Distribution ◽  
Weighted Sum ◽  
Two Dimensional ◽  
Sum Of Digits ◽  
Higher Dimensional ◽  
Distribution Modulo One ◽  
Sum Of Digits Function ◽  
Uniform Distribution Modulo One ◽  
Van Der Corput Sequence ◽  
Distribution Modulo

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.

scholarly journals ON INTEGER SEQUENCES GENERATED BY LINEAR MAPS

2009 ◽  
Vol 51 (2) ◽  
pp. 243-252
Author(s):  
ARTŪRAS DUBICKAS
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Real Number ◽  
Constant Independent ◽  
Integer Sequences ◽  
Real Numbers ◽  
Complexity Function ◽  
Linear Maps ◽  
Coprime Integers ◽  
Positive Integers

AbstractLetx0<x1<x2< ⋅⋅⋅ be an increasing sequence of positive integers given by the formulaxn=⌊βxn−1+ γ⌋ forn=1, 2, 3, . . ., where β > 1 and γ are real numbers andx0is a positive integer. We describe the conditions on integersbd, . . .,b0, not all zero, and on a real number β > 1 under which the sequence of integerswn=bdxn+d+ ⋅⋅⋅ +b0xn,n=0, 1, 2, . . ., is bounded by a constant independent ofn. The conditions under which this sequence can be ultimately periodic are also described. Finally, we prove a lower bound on the complexity function of the sequenceqxn+1−pxn∈ {0, 1, . . .,q−1},n=0, 1, 2, . . ., wherex0is a positive integer,p>q> 1 are coprime integers andxn=⌈pxn−1/q⌉ forn=1, 2, 3, . . . A similar speculative result concerning the complexity of the sequence of alternatives (F:x↦x/2 orS:x↦(3x+1)/2) in the 3x+1 problem is also given.

scholarly journals A note on the lower bound of representation functions

2021 ◽  
pp. 1-8
Author(s):  
Xing-Wang Jiang ◽  
Csaba Sándor ◽  
Quan-Hui Yang
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Number Of Solutions

For a set [Formula: see text] of nonnegative integers, let [Formula: see text] denote the number of solutions to [Formula: see text] with [Formula: see text], [Formula: see text]. Let [Formula: see text] be the Thue–Morse sequence and [Formula: see text]. Let [Formula: see text] and [Formula: see text] be a positive integer such that [Formula: see text] for all [Formula: see text]. Previously, the first author proved that if [Formula: see text] and [Formula: see text], then [Formula: see text] for all [Formula: see text]. In this paper, we prove that the above lower bound is nearly best possible. We also get some other results.

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