Gregory A. Freiman

On two additive problems

Denote by A a set of x different natural numbers. The following two results are obtained: 1. (1) For sufficiently large x and A ⊂ [1, 3x − 3] there exists a subset B ⊂ A such that for some l ∈ Z ∑aiϵB ai=2l 2. (2) For sufficiently large x and A ⊂ [1, 4x − 4] there exists a subset B ⊂ A such that Σai ∈ B ai is a square-free number.

Asymptotic formula for a partition function of reversible coagulation -fragmentation processes.

We construct a probability model seemingly unrelated to the considered stochastic process of coagulation and fragmentation. By proving for this model the local limit theorem, we establish the asymptotic formula for the partition function of the equilibrium measure for a wide class of parameter functions of the process. This formula proves the conjecture stated in [5] for the above class of processes. The method used goes back to A. Khintchine.

Solving dense subset-sum problems by using analytical number theory

Abstract Theorems from analytical number theory are used to derive new algorithms for the subset-sum problem. The algorithms work for a large number of variables (m) with values that are bounded above. The bound (l) depends moderately on m. While the dynamic programming approach yields an O(lm2) algorithm, the new algorithms are substantially faster.

On sums of subsets of a set of integers

AbstractForr≧2 letp(n, r) denote the maximum cardinality of a subsetA ofN={1, 2,...,n} such that there are noB⊂A and an integery with

On a product of finite subsets in a torsion-free group

\mathop \sum \limits_{b \in B} b = y^r

A method to suppress high peaks in BPSK-modulated OFDM signal

b=yr. It is shown that for anyε>0 andn>n(ε), (1+o(1))21/(r+1)n(r−1)/(r+1)≦p(n, r)≦nɛ+2/3 for allr≦5, and that for every fixedr≧6,p(n, r)=(1+o(1))·21/(r+1)n(r−1)/(r+1) asn→∞. Letf(n, m) denote the maximum cardinality of a subsetA ofN such that there is noB⊂A the sum of whose elements ism. It is proved that for 3n6/3+ɛ≦m≦n2/20 log2n andn>n(ε), f(n, m)=[n/s]+s−2, wheres is the smallest integer that does not dividem. A special case of this result establishes a conjecture of Erdős and Graham.

On Bounds for the Concentration Function II

Abstract An analytical method is developed to prove that, for the integer set Aϵ[1,l], with l>;l0 and |A|=m>;c 1 l 1 2 (log l) 1 2 , the set A∗ of subset sums contains a long arithmetic progression of length larger than c2m2. Here l0 and c1 are sufficiently large constants and c2 is some positive constant. This result gives a possibility to solve new algorithmic and combinatorial problems connected with subset sums.

Clustering in coagulation - fragmentation processes, random combinatorial structures and additive number systems: Asymptotic formulae and limiting laws.

Let G be a torsion-free group, and K and A4 finite subsets of G with ) Kj >, 2 and JMJ > 2. We denote by KM the set of the elements g E G which have at least one representation of the form g = km, where kE K and rng M. It is known [Ke] that in the described situation the following inequality always holds: IKMI 2 IK:) + /MI 1. (1.1) The purpose of this paper is to determine the structure of K and M when the order of KM happens to be the minimal possible, namely, when 1 KM1 = 1 KI + I MI 1. We prove the following