Minimal bases and maximal nonbases in additive number theory
نویسندگان
چکیده
منابع مشابه
Systems of Distinct Representatives and Minimal Bases in Additive Number Theory
The set A of nonnegative integers is an asymptotic basis of order h if every sufficiently large integer is the sum of h elements of A . For example, the squares form an asymptotic basis of order 4 and the square-free numbers form an asymptotic basis of order 2 . If A is an asymptotic basis of order h, but no proper subset of A is an asymptotic basis of order h, then A is a minimal asymptotic ba...
متن کاملNonbases of Density Zero Not Contained in Maximal Nonbases
A sequence A = {a t } of non-negative integers is a basis if every sufficiently large integer n can be written in the form n = a t+aj with a,, aj e A . If A is not a basis, then A is called a nonbasis . The nonbasis A is maximal if A u {b} is a basis for every b 0 A . We construct a nonbasis A of density zero, in particular, with A(x) = O(V x), such that A cannot be imbedded as a subset of any ...
متن کاملBases and Nonbases of Square-Free Integers
A basis is a set A of nonnegative integers such that every sufficiently large integer n can be represented in the form n = a + a i with a , ai e A . If A is a basis, but no proper subset of A is a basis, then A is a minimal basis . A nonbasis is a set of nonnegative integers that is not a basis, and a nonbasis A is maximal if every proper superset of A is a basis . In this paper, minimal bases ...
متن کاملSupersequences, Rearrangements of Sequences, and the Spectrum of Bases in Additive Number Theory
The set A = {an}∞n=1 of nonnegative integers is an asymptotic basis of order h if every sufficiently large integer can be represented as the sum of h elements of A. If an ∼ αn for some real number α > 0, then α is called an additive eigenvalue of order h. The additive spectrum of order h is the set N (h) consisting of all additive eigenvalues of order h. It is proved that there is a positive nu...
متن کاملSome Problems in Additive Number Theory
(3) f(x) = (log x/log 2) + 0(1)? 1\Mloser and I asked : Is it true that f(2 11) >_ k+2 for sufficiently large k? Conway and Guy showed that the answer is affirmative (unpublished) . P. Erdös, Problems and results in additive number theory, Colloque, Théorie des Nombres, Bruxelles 1955, p . 137 . 2. Let 1 < a 1< . . . < ak <_ x be a sequence of integers so that all the sums ai,+ . . .+ais, i 1 <...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 1974
ISSN: 0022-314X
DOI: 10.1016/0022-314x(74)90028-6