On a problem in additive number theory

For a set A, let P(A) be the set of all finite subset sums of A. We prove that if a sequence B={b ₁<b ₂<⋯} of integers satisfies b ₁≧11 and b _n+1≧3b _n +5 (n=1,2,…), then there exists a sequence of positive integers A={a ₁<a ₂<⋯} for which P(A)=ℕ∖B. On the other hand, if a sequence B={b ₁<b ₂<⋯} of positive integers satisfies either b ₁=10 or b ₂=3b ₁+4, then there is no sequence A of positive integers for which P(A)=ℕ∖B.     

A lattice point problem and additive number theory

For every dimensiond1 there exists a constantc=c(d) such that for alln1, every set of at leastcn lattice points in thed-dimensional Euclidean space contains a subset of cardinality preciselyn whose centroid is also a lattice point. The proof combines techniques from additive number theory with results about the expansion properties of Cayley graphs with given eigenvalues.     

Generalization of the additive problem of divisors

An asymptotic formula is obtained for a sum of the form _{m f (m) (n–m) where f(m) is a multiplicative function with additional constraints, and (m) is the number of divisors of m.Translated from Matematicheskie Zametki, Vol. 7, No. 4, pp. 477–482, April, 1970.}     

An additive divisor problem with a growing number of factors

Let τk(n) be the number of representations ofn as the product ofk positive factors, τ(n)=τ(n). The asymptotics of Σ _n≤x τ _k (n)τ(n+1) for 80k ¹⁰ (lnlnx)³≤lnx is shown to be uniform with respect tok. Translated fromMatematicheskie Zametki, Vol. 61, No. 3, pp. 391–406, March, 1997. Translated by N. K. Kulman     

Promoting number theory in high schools or birthday problem and number theory

The author introduces the birthday problem in this article. This can amuse willing members of any birthday party. This problem can also be used as the motivational first day lecture in number theory for the gifted students in high schools or in community colleges or in undergraduate classes in colleges.     

Minimal bases and maximal nonbases in additive number theory

An asymptotic basis $\text{A}$ of order h is minimal if no proper subset of $\text{A}$ is an asymptotic basis of order h. Examples are constructed of minimal asymptotic bases, and also of an asymptotic basis of order two no subset of which is minimal.If $\text{B}$ is a set of nonnegative integers which is not a basis (resp. asymptotic basis) of order h, but such that every proper superset of $\text{B}$ is a basis (resp. asymptotic basis) of order h, then $\text{B}$ is a maximal nonbasis (resp. maximal asymptotic nonbasis) of order h. Examples of such sets are constructed, and it is proved that every set not a basis of order h is a subset of a maximal nonbasis of order h.     

Probabilistic number theory in additive arithmetic semigroups II

We prove two quantitative mean-value theorems of completely multiplicative functions on additive arithmetic semigroups. On the basis of the two theorems, a central limit theorem of additive functions on additive arithmetic semigroups is proved with a best possible error estimate. This generalizes the vital results of Halász and Elliott in classical probabilistic number theory to function fields. Received October 26, 1998; in final form April 5, 2000 / Published online October 11, 2000     

Direct and inverse problems in additive number theory and in non-abelian group theory

We obtain new direct and inverse results for Minkowski sums of dilates and we apply them to solve certain direct and inverse problems in Baumslag–Solitar groups, assuming appropriate small doubling properties.     

Generalization of the problem of the number of partitions of a finite set

A unified scheme is proposed for obtaining the generating functions for combinatorial objects defined on partitions of finite sets.Translated from Matematicheskie Zametki, Vol. 10, No. 3, pp. 361–367, September, 1971.     

On an extremal problem in number theory

We define h(n) to be the largest function of n such that from any set of n nonzero integers, one can always extract a subset of h(n) integers with the property that any two sums formed from its elements are equal only if they have equal number of summands. A result of Erdös implies that $\text{h(n) ? n}{}^{\mathrm{<mtext>1</mtext><mtext>3</mtext>}}$, and it is the aim of the present paper to obtain the refinement $\text{h(n) ? n}{}^{\mathrm{<mtext>1</mtext><mtext>3</mtext>}}\text{(}\text{log}\text{n)}{}^{\mathrm{<mtext>1</mtext><mtext>3</mtext>}}$.