On additive representation of integers

Acta Mathematica Hungarica -     

On additive partitions of integers

Given a linear recurrence integer sequence U = {u_n}, u_n+2 = u_n+1 + u_r, n ? 1, u₁ = 1, u₂> u₁, we prove that the set of positive integers can be partitioned uniquely into two disjoint subsets such that the sum of any two distinct members from any one set can never be in U. We give a graph theoretic interpretation of this result, study related problems and discuss possible generalizations.     

On the additive completion of sets of integers

Denote by k = k(N) the least integer for which there exists integers b₁, b₂, …, b_k satisfying 0 ≤ b₁ ≤ b₂ ≤ … ≤ b_k ≤ N such that every integer in |1, N| can be written in the form i² + b_j. It is shown that for all sufficiently large N, k ≥ (1.147)√N.     

On an additive property of sequences of nonnegative integers

Periodica Mathematica Hungarica - Let a1&;lt;... be an infinite sequence of positive integers, let k≥2 be a fixed integer and denote by Rk(n) the number of solutions of n=ai1+ai2+...+aik....     

On the structure of large sum-free sets of integers

A set of integers is called sum-free if it contains no triple (x, y, z) of not necessarily distinct elements with x + y = z. In this paper, we provide a structural characterisation of sum-free subsets of {1, 2,..., n} of density at least 2/5 ? c, where c is an absolute positive constant. As an application, we derive a stability version of Hu’s Theorem [Proc. Amer. Math. Soc. 80 (1980), 711–712] about the maximum size of a union of two sum-free sets in {1, 2,..., n}. We then use this result to show that the number of subsets of {1, 2,..., n} which can be partitioned into two sum-free sets is Θ(2^4n/5), confirming a conjecture of Hancock, Staden and Treglown [arXiv:1701.04754].     

An additive problem about powers of fixed integers

Résumé  SoitA un ensemble fini d'entiers ≥2. Nous étudions les propriétés de l'ensemble Σ(Pow(A)) des entiers positifs qui sont une somme de puissances distinctes d'éléments deA. Erd?s posa le problème suivant: démontrer que Σ(Pow({3,4})) a densité asymtotique superieure positive. Nous démontrons que la fonction qui les énumère vérifieP _{3,4}(x)≫x^0.9659.      

The representation of integers by binary additive forms

Let a, b and n be integers with 3. We show that, in the sense of natural density, almost all integers represented by the binary form axⁿ – byⁿ are thus represented essentially uniquely. By exploiting this conclusion, we derive an asymptotic formula for the total number of integers represented by such a form. These conclusions augment earlier work of Hooley concerning binary cubic and quartic forms, and generalise or sharpen work of Hooley, Greaves, and Skinner and Wooley concerning sums and differences of two nth powers.     

On the structure of a random sum-free set of positive integers

Cameron introduced a natural probability measure on the set of sum-free sets, and asked which sets of sum-free sets have a positive probability of occurring in this probability measure. He showed that the set of subsets of the odd numbers has a positive probability, and that the set of subsets of any sum-free set corresponding to a complete modular sum-free set also has a positive probability of occurring. In this paper we consider, for every sum-free set S, the representation function r_s(n), and show that if r_s(n) grows sufficiently quickly then the set of subsets of S has positive probability, and conversely, that if r_s(n) has a sub-sequence with suitably slow growth, then the set of subsets of S has probability zero. The results include those of Cameron mentioned above as particular cases.     

On the additive cyclic structure of quasi-cyclic codes

An index $?$, length $m?$ quasi-cyclic code can be viewed as a cyclic code of length $m$ over the field ${\mathbb{F}}_{q^{?}}$ via a basis of the extension ${\mathbb{F}}_{q^{?}}∕{\mathbb{F}}_q$. However, this cyclic code is only linear over ${\mathbb{F}}_q$, making it an additive cyclic code, or an ${\mathbb{F}}_q$-linear cyclic code, over the alphabet ${\mathbb{F}}_{q^{?}}$. This approach was recently used in Shi et al. (2017) [16] to study a class of quasi-cyclic codes, and more importantly in Shi et al. (2017) [17] to settle a long-standing question on the asymptotic performance of cyclic codes. Here, we answer one of the problems posed in these two articles, and characterize those quasi-cyclic codes which have ${\mathbb{F}}_{q^{?}}$-linear cyclic images under a basis of the extension ${\mathbb{F}}_{q^{?}}∕{\mathbb{F}}_q$. Our characterizations are based on the module structure of quasi-cyclic codes, as well as on their CRT decompositions into constituents. In the case of a polynomial basis, we characterize the constituents by using the theory of invariant subspaces of operators. We also observe that analogous results extend to the case of quasi-twisted codes.     

Noncanonical number systems in the integers

The well-known binary and decimal representations of the integers, and other similar number systems, admit many generalisations. Here, we investigate whether still every integer could have a finite expansion on a given integer base b, when we choose a digit set that does not contain 0. We prove that such digit sets exist and we provide infinitely many examples for every base b with |b|?4, and for b=−2. For the special case b=−2, we give a full characterisation of all valid digit sets.     

Solving polynomial systems in integers

The paper describes a method for determining integer solutions of a homogeneous polynomial system with integer coefficients which has finitely many solutions in the projective space over the field of complex numbers under the assumption that these solutions have a certain property.     

On products of integers

The main objective of this paper is to investigate the relation between the number of integers in a given subset $\text{A}$ of the integers 1, 2,…, n and the number of integers that can be chosen from 1, 2,…, n so that their pairwise products all appear in $\text{A}$. Other related problems are also considered.     

On the existence of a mixed sum of additive systems

A representation is obtained for a mixed sum of additive systems with values in a Banach ringX with identity and norm. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 6, pp. 699–707, June, 1994.     

On exact coverings of the integers

By an exact covering of modulusm, we mean a finite set of liner congruencesx≡a _i (modm _i ), (i=1,2,...r) with the properties: (I)m _i ∣m, (i=1,2,...,r); (II) Each integer satisfies precisely one of the congruences. Let α≥0, β≥0, be integers and letp andq be primes. Let μ (m) senote the Möbius function. Letm=p ^α q ^β and letT(m) be the number of exact coverings of modulusm. Then,T(m) is given recursively by $$\mathop \Sigma \limits_{d/m} \mu (d)\left( {T\left( {\frac{m}{d}} \right)} \right)^d = 1$$ .     

On the distribution of square-full integers

Let A(x) A(x) be the number of square-full integers \leqq x \leqq x and let D(x) \Delta(x) be the error term in the asymptotic formula for A(x) A(x) . Under the Riemann hypothesis, we show that D(x) << x^[12/85]+e \Delta(x)\ll x^{{12\over 85}+\varepsilon} . This improves the earlier results of Zhu and Yu [17], Cao [4, II], Liu [9] and Wu [16], which requires [ 1/7 ] 1\over 7 in place of [ 12/85 ] 12\over 85 .