scholarly journals The Tu–Deng Conjecture holds Almost Surely

10.37236/7178 ◽  
2019 ◽  
Vol 26 (1) ◽  
Author(s):  
Lukas Spiegelhofer ◽  
Michael Wallner
Keyword(s):  
Positive Integer ◽  
Hamming Weight ◽  
Binary Expansion ◽  
Sum Of Digits

The Tu–Deng Conjecture is concerned with the sum of digits $w(n)$ of $n$ in base $2$ (the Hamming weight of the binary expansion of $n$) and states the following: assume that $k$ is a positive integer and$1\leqslant t<2^k-1$. Then\[\Bigl \lvert\Bigl\{(a,b)\in\bigl\{0,\ldots,2^k-2\bigr\}^2:a+b\equiv t\bmod 2^k-1, w(a)+w(b)<k\Bigr\}\Bigr \rvert\leqslant 2^{k-1}.\]We prove that the Tu–Deng Conjecture holds almost surely in the following sense: the proportion of $t\in[1,2^k-2]$ such that the above inequality holds approaches $1$ as $k\rightarrow\infty$.Moreover, we prove that the Tu–Deng Conjecture implies a conjecture due to T. W. Cusick concerning the sum of digits of $n$ and $n+t$.

scholarly journals Statistical independence in mathematics–the key to a Gaussian law

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.

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 On the sum of digits of some sequences of integers

2013 ◽  
Vol 11 (1) ◽  
Author(s):  
Javier Cilleruelo ◽  
Florian Luca ◽  
Juanjo Rué ◽  
Ana Zumalacárregui
Keyword(s):  
Number Theory ◽  
Positive Integer ◽  
Integer Sequences ◽  
Sum Of Digits ◽  
Fixed Positive Integer ◽  
Almost All

AbstractLet b ≥ 2 be a fixed positive integer. We show for a wide variety of sequences {a n}n=1∞ that for almost all n the sum of digits of a n in base b is at least c b log n, where c b is a constant depending on b and on the sequence. Our approach covers several integer sequences arising from number theory and combinatorics.

Sequential motion planning algorithms in real projective spaces: An approach to their immersion dimension

2018 ◽  
Vol 30 (2) ◽  
pp. 397-417 ◽  
Author(s):  
Natalia Cadavid-Aguilar ◽  
Jesús González ◽  
Darwin Gutiérrez ◽  
Aldo Guzmán-Sáenz ◽  
Adriana Lara
Keyword(s):  
Positive Integer ◽  
Projective Space ◽  
Motion Planning ◽  
Product Space ◽  
Projective Spaces ◽  
Topological Complexity ◽  
Real Projective Space ◽  
Binary Expansion ◽  
The Real

AbstractThes-th higher topological complexity{\operatorname{TC}_{s}(X)}of a spaceXcan be estimated from above by homotopical methods, and from below by homological methods. We give a thorough analysis of the gap between such estimates when{X=\operatorname{\mathbb{R}P}^{m}}, the real projective space of dimensionm. In particular, we describe a number{r(m)}, which depends on the structure of zeros and ones in the binary expansion ofm, and with the property that{0\leq sm-\operatorname{TC}_{s}(\operatorname{\mathbb{R}P}^{m})\leq\delta_{s}(% m)}for{s\geq r(m)}, where{\delta_{s}(m)=(0,1,0)}for{m\equiv(0,1,2)\bmod 4}. Such an estimation for{\operatorname{TC}_{s}(\operatorname{\mathbb{R}P}^{m})}appears to be closely related to the determination of the Euclidean immersion dimension of{\operatorname{\mathbb{R}P}^{m}}. We illustrate the phenomenon in the case{m=3\cdot 2^{a}}. In addition, we show that, for large enoughsand evenm,{\operatorname{TC}_{s}(\operatorname{\mathbb{R}P}^{m})}is characterized as the smallest positive integer{t=t(m,s)}for which there is a suitable equivariant map from Davis’ projective product space{\mathrm{P}_{\mathbf{m}_{s}}}to the{(t+1)}-st join-power{((\mathbb{Z}_{2})^{s-1})^{\ast(t+1)}}. This is a (partial, but conjecturally complete) generalization of the work of Farber, Tabachnikov and Yuzvinsky relating{\operatorname{TC}_{2}}to the immersion dimension of real projective spaces.

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 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 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.

The Order of Monochromatic Subgraphs with a Given Minimum Degree

10.37236/1725 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Yair Caro ◽  
Raphael Yuster
Keyword(s):  
Positive Integer ◽  
Minimum Degree

Let $G$ be a graph. For a given positive integer $d$, let $f_G(d)$ denote the largest integer $t$ such that in every coloring of the edges of $G$ with two colors there is a monochromatic subgraph with minimum degree at least $d$ and order at least $t$. Let $f_G(d)=0$ in case there is a $2$-coloring of the edges of $G$ with no such monochromatic subgraph. Let $f(n,k,d)$ denote the minimum of $f_G(d)$ where $G$ ranges over all graphs with $n$ vertices and minimum degree at least $k$. In this paper we establish $f(n,k,d)$ whenever $k$ or $n-k$ are fixed, and $n$ is sufficiently large. We also consider the case where more than two colors are allowed.

Extensions of Rings Having McCoy Condition

2009 ◽  
Vol 52 (2) ◽  
pp. 267-272 ◽  
Author(s):  
Muhammet Tamer Koşan
Keyword(s):  
Positive Integer ◽  
Associative Ring ◽  
Nonzero Element ◽  
Ring Extensions

AbstractLet R be an associative ring with unity. Then R is said to be a right McCoy ring when the equation f (x)g(x) = 0 (over R[x]), where 0 ≠ f (x), g(x) ∈ R[x], implies that there exists a nonzero element c ∈ R such that f (x)c = 0. In this paper, we characterize some basic ring extensions of right McCoy rings and we prove that if R is a right McCoy ring, then R[x]/(xn) is a right McCoy ring for any positive integer n ≥ 2.

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