scholarly journals When is an automatic set an additive basis?

2018 ◽  
Vol 5 (6) ◽  
pp. 50-63 ◽  
Author(s):  
Jason Bell ◽  
Kathryn Hare ◽  
Jeffrey Shallit
Keyword(s):  
Additive Basis

Additive bases via Fourier analysis

2021 ◽  
pp. 1-12
Author(s):  
Bodan Arsovski
Keyword(s):  
Abelian Group ◽  
Fourier Analysis ◽  
Vector Space ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Probabilistic Method ◽  
Additive Basis ◽  
Generating Sets

Abstract Extending a result by Alon, Linial, and Meshulam to abelian groups, we prove that if G is a finite abelian group of exponent m and S is a sequence of elements of G such that any subsequence of S consisting of at least $$|S| - m\ln |G|$$ elements generates G, then S is an additive basis of G . We also prove that the additive span of any l generating sets of G contains a coset of a subgroup of size at least $$|G{|^{1 - c{ \in ^l}}}$$ for certain c=c(m) and $$ \in = \in (m) < 1$$ ; we use the probabilistic method to give sharper values of c(m) and $$ \in (m)$$ in the case when G is a vector space; and we give new proofs of related known results.

scholarly journals ADDITIVE BASES AND NIVEN NUMBERS

2021 ◽  
pp. 1-8
Author(s):  
CARLO SANNA
Keyword(s):  
Positive Integer ◽  
Natural Number ◽  
Riemann Hypothesis ◽  
Additive Basis

Abstract Let $g \geq 2$ be an integer. A natural number is said to be a base-g Niven number if it is divisible by the sum of its base-g digits. Assuming Hooley’s Riemann hypothesis, we prove that the set of base-g Niven numbers is an additive basis, that is, there exists a positive integer $C_g$ such that every natural number is the sum of at most $C_g$ base-g Niven numbers.

An inverse theorem for additive bases

2016 ◽  
Vol 12 (06) ◽  
pp. 1509-1518 ◽  
Author(s):  
Yongke Qu ◽  
Dongchun Han
Keyword(s):  
Abelian Group ◽  
Proper Subgroup ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Regular Sequence ◽  
Inverse Theorem ◽  
Additive Basis

Let [Formula: see text] be a finite abelian group of order [Formula: see text], and [Formula: see text] be the smallest prime dividing [Formula: see text]. Let [Formula: see text] be a sequence over [Formula: see text]. We say that [Formula: see text] is regular if for every proper subgroup [Formula: see text], [Formula: see text] contains at most [Formula: see text] terms from [Formula: see text]. Let [Formula: see text] be the smallest integer [Formula: see text] such that every regular sequence [Formula: see text] over [Formula: see text] of length [Formula: see text] forms an additive basis of [Formula: see text], i.e. [Formula: see text]. Recently, [Formula: see text] was determined for many abelian groups. In this paper, we determined [Formula: see text] for more abelian groups and characterize the structure of the regular sequence [Formula: see text] over [Formula: see text] of length [Formula: see text] and [Formula: see text].

Additive bases of Cp ⊕ Cpn

2017 ◽  
Vol 13 (09) ◽  
pp. 2453-2459
Author(s):  
Yongke Qu ◽  
Dongchun Han
Keyword(s):  
Abelian Group ◽  
Proper Subgroup ◽  
Finite Abelian Group ◽  
Regular Sequence ◽  
Additive Basis

Let [Formula: see text] be a finite abelian group, and [Formula: see text] be the smallest prime dividing [Formula: see text]. Let [Formula: see text] be a sequence over [Formula: see text]. We say that [Formula: see text] is regular if for every proper subgroup [Formula: see text], [Formula: see text] contains at most [Formula: see text] terms from [Formula: see text]. Let [Formula: see text] be the smallest integer [Formula: see text] such that every regular sequence [Formula: see text] over [Formula: see text] of length [Formula: see text] forms an additive basis of [Formula: see text], i.e. [Formula: see text]. In this paper, we show that [Formula: see text] where [Formula: see text].

scholarly journals FINITELY STABLE ADDITIVE BASES

2018 ◽  
Vol 97 (3) ◽  
pp. 360-362
Author(s):  
L. A. FERREIRA
Keyword(s):  
Sufficient Condition ◽  
Additive Basis

An additive basis $A$ is finitely stable when the order of $A$ is equal to the order of $A\cup F$ for all finite subsets $F\subseteq \mathbb{N}$. We give a sufficient condition for an additive basis to be finitely stable. In particular, we prove that $\mathbb{N}^{2}$ is finitely stable.

ON THE CRITICAL NUMBER OF FINITE GROUPS OF ORDER pq

2012 ◽  
Vol 08 (05) ◽  
pp. 1271-1279 ◽  
Author(s):  
Q. H. WANG ◽  
J. J. ZHUANG
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Critical Number ◽  
Additive Basis ◽  
Empty Subset ◽  
A Finite Group

Let G be a finite group and S be a subset of G\{0}. We call S an additive basis of G if every element of G can be expressed as a sum over a non-empty subset in some order. Let cr (G) be the smallest integer t such that every subset of G\{0} of cardinality t forms an additive basis of G. In this paper, we determine cr (G) for all groups G of order pq, where p, q are primes.

scholarly journals ADDITIVE BASES IN ABELIAN GROUPS

2010 ◽  
Vol 06 (04) ◽  
pp. 799-809
Author(s):  
VSEVOLOD F. LEV ◽  
MIKHAIL E. MUZYCHUK ◽  
ROM PINCHASI
Keyword(s):  
Abelian Group ◽  
Lower Bound ◽  
Vector Space ◽  
Abelian Groups ◽  
Unit Cube ◽  
Strong Form ◽  
Additive Basis

Let G be a finite, non-trivial Abelian group of exponent m, and suppose that B1,…, Bk are generating subsets of G. We prove that if k > 2m ln log2 |G|, then the multiset union B1 ∪ Bk forms an additive basis of G; that is, for every g ∈ G, there exist A1 ⊆ B1,…, Ak ⊆ Bk such that [Formula: see text]. This generalizes a result of Alon, Linial and Meshulam on the additive bases conjecture. As another step towards proving the conjecture, in the case where B1,…, Bk are finite subsets of a vector space, we obtain lower-bound estimates for the number of distinct values, attained by the sums of the form [Formula: see text], where Ai vary over all subsets of Bi for each i = 1,…, k. Finally, we establish a surprising relation between the additive bases conjecture and the problem of covering the vertices of a unit cube by translates of a lattice, and present a reformulation of (the strong form of) the conjecture in terms of coverings.

scholarly journals (–1, 1) metabelian rings

2017 ◽  
Vol 35 (2) ◽  
pp. 115-125
Author(s):  
Karamsi Jayalakshmi ◽  
Kommaddi Hari Babu
Keyword(s):  
Additive Basis ◽  
Torsion Free

The structure of the set of all non- nilpotent (–1, 1) metabelian rings is studied. An additive basis of a free (–1, 1) metabelian ring is constructed. It is proved that any identity in a non- nilpotent 2, 3-torsion free (–1, 1) metabelian ring of degree greater than or equal to 6 is a consequence four defining identities of  m where m is the metabelian (–1, 1) ring.

VARIATIONS ON A THEME OF CASSELS FOR ADDITIVE BASES

2006 ◽  
Vol 02 (02) ◽  
pp. 249-265 ◽  
Author(s):  
G. GREKOS ◽  
L. HADDAD ◽  
C. HELOU ◽  
J. PIHKO
Keyword(s):  
Large Class ◽  
Natural Numbers ◽  
Average Order ◽  
Basic Properties ◽  
Additive Basis ◽  
Perfect Squares ◽  
Variations On A Theme

We introduce the notion of caliber, cal (A, B), of a strictly increasing sequence of natural numbers A with respect to another one B, as the limit inferior of the ratio of the nth term of A to that of B. We further consider the limit superior t(A) of the average order of the number of representations of an integer as a sum of two elements of A. We give some basic properties of each notion and we relate the two together, thus yielding a generalization, of the form t(A) ≤ t(B)/ cal (A, B), of a result of Cassels specific to the case where A is an additive basis of the natural numbers and B is the sequence of perfect squares. We also provide some formulas for the computation of t(A) in a large class of cases, and give some examples.

Additive basis for multivector information

2007 ◽  
Author(s):  
Victor I. Tarkhanov ◽  
Armand Ebanga
Keyword(s):  
Additive Basis
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