Problems In Combinatorial Number Theory Research Articles

On a communication complexity problem in combinatorial number theory

The original knapsack problem is well known to be NP-complete. In a multidimensional version one have to decide whether a $p\in \N^k$ is in a sumset-sum of a set $X \subseteq \N^k$ or not. In this paper we are going to investigate a communication complexity problem related to this.

A genetic algorithm for the minimum generating set problem

Given a set of positive integers S, the minimum generating set problem consists in finding a set of positive integers T with a minimum cardinality such that every element of S can be expressed as the sum of a subset of elements in T. It constitutes a natural problem in combinatorial number theory and is related to some real-world problems, such as planning radiation therapies.We present a new formulation to this problem (based on the terminology for the multiple knapsack problem) that is used to design an evolutionary approach whose performance is driven by three search strategies; a novel random greedy heuristic scheme that is employed to construct initial solutions, a specialized crossover operator inspired by real-parameter crossovers and a restart mechanism that is incorporated to avoid premature convergence. Computational results for problem instances involving up to 100,000 elements show that our innovative genetic algorithm is a very attractive alternative to the existing approaches.

Nonstandard analysis and its applications

In this paper, we briefly introduce the foundation of nonstandard analysis and its applications to other fields of mathematics, especially to combinatorial number theory. The foundation part includes the basic knowledge of mathematical logic, the construction of nonstandard structures, and some principles and properties commonly used in nonstandard analysis. The application part includes theorems on the existence of strong solutions for some stochastic differential equations, solution to the Hilberts fifth problem for local groups, exact law of large numbers and its applications to mathematical economics, and sumset phenomenon, Plunnecke type inequalities for densities, and Freimans inverse problems in combinatorial number theory.

Computer-aided proof of Erdős discrepancy properties

In 1930s Paul Erdős conjectured that for any positive integer C in any infinite ±1 sequence (xn) there exists a subsequence xd,x2d,x3d,…,xkd, for some positive integers k and d, such that |∑i=1kxi⋅d|>C. The conjecture has been referred to as one of the major open problems in combinatorial number theory and discrepancy theory. For the particular case of C=1 a human proof of the conjecture exists; for C=2 a bespoke computer program had generated sequences of length 1124 of discrepancy 2, but the status of the conjecture remained open even for such a small bound. We show that by encoding the problem into Boolean satisfiability and applying the state of the art SAT solvers, one can obtain a discrepancy 2 sequence of length 1160 and a proof of the Erdős discrepancy conjecture for C=2, claiming that no discrepancy 2 sequence of length 1161, or more, exists. In the similar way, we obtain a precise bound of 127 645 on the maximal lengths of both multiplicative and completely multiplicative sequences of discrepancy 3. We also demonstrate that unrestricted discrepancy 3 sequences can be longer than 130 000.

A nonstandard technique in combinatorial number theory

In Di Nasso (2015) and Luperi Baglini (2012) it has been introduced a technique, based on nonstandard analysis, to study some problems in combinatorial number theory. In this paper we review such a technique and we present three of its applications: the first one is a new proof of a known result regarding the algebra of βN, namely that the center of the semigroup (βN,⊕) is N; the second one is a generalization of a theorem of Bergelson and Hindman on arithmetic progressions of length three; the third one regards the study of which polynomials in several variables with integers coefficients have a monochromatic solution for every finite coloring of N. We will study this last application in more detail: we will prove some algebraical properties of the set P of such polynomials and we will present a few examples of nonlinear polynomials in P.In the first part of the paper we will recall the main results of the nonstandard technique that we want to use, which is based on a characterization of ultrafilters by means of nonstandard analysis.

Extremal problems on sum-free sets and coverings in tridimensional spaces

Given a prime power q, define c(q) as the minimum cardinality of a subset H of $${\mathbb{F}}^3_q$$ which satisfies the following property: every vector in this space differs in at most 1 coordinate from a multiple of a vector in H. In this work, we introduce two extremal problems in combinatorial number theory aiming to discuss a known connection between the corresponding coverings and sum-free sets. Also, we provide several bounds on these maps which yield new classes of coverings, improving the previous upper bound on c(q).

FINITE GROUPS WITH AN AUTOMORPHISM CUBING A LARGE FRACTION OF ELEMENTS

We investigate the possible structures imposed on a finite group by its possession of an automorphism sending a large fraction of the group elements to their cubes, the philosophy being that this should force the group to be, in some sense, close to abelian. We prove two main theorems. In the first, we completely classify all finite groups with an automorphism cubing more than half their elements. All such groups are either nilpotent class 2 or possess an abelian subgroup of index 2. For our second theorem, we show that if a group possesses an automorphism sending more than 4/15 of its elements to their cubes, then it must be solvable. The group A_5 shows that this result is best-possible. Both our main findings closely parallel results of previous authors on finite groups possessing an automorphism which inverts many elements. The technicalities of the new proofs are somewhat more subtle, and also throw up a nice connection to a basic problem in combinatorial number theory, namely the study of subsets of finite cyclic groups which avoid non-trivial solutions to one or more translation invariant linear equations.

Parity check matrices and product representations of squares

Let $$ N_{\mathbb{F}} $$ (n,k,r) denote the maximum number of columns in an n-row matrix with entries in a finite field $$ \mathbb{F} $$ in which each column has at most r nonzero entries and every k columns are linearly independent over $$ \mathbb{F} $$ . We obtain near-optimal upper bounds for $$ N_{\mathbb{F}} $$ (n,k,r) in the case k > r. Namely, we show that $$ N_\mathbb{F} (n,k,r) \ll n^{\frac{r} {2} + \frac{{cr}} {k}} $$ where $$ c \approx \frac{4} {3} $$ for large k. Our method is based on a novel reduction of the problem to the extremal problem for cycles in graphs, and yields a fast algorithm for finding short linear dependencies. We present additional applications of this method to a problem on hypergraphs and a problem in combinatorial number theory.

Counting primes, groups, and manifolds.

Let Lambda=SL(2)(Z) be the modular group and let c(n)(Lambda) be the number of congruence subgroups of Lambda of index at most n. We prove that lim(n--> infinity )(log c(n)(Lambda)/((log n)(2)/log log n))=(3-2(sqrt)2)/4. The proof is based on the Bombieri-Vinogradov "Riemann hypothesis on the average" and on the solution of a new type of extremal problem in combinatorial number theory. Similar surprisingly sharp estimates are obtained for the subgroup growth of lattices in higher rank semisimple Lie groups. If G is such a Lie group and Gamma is an irreducible lattice of G it turns out that the subgroup growth of Gamma is independent of the lattice and depends only on the Lie type of the direct factors of G. It can be calculated easily from the root system. The most general case of this result relies on the Generalized Riemann Hypothesis, but many special cases are unconditional. The proofs use techniques from number theory, algebraic groups, finite group theory, and combinatorics.

Numerical estimation of the asymptotic behaviour of solid partitions of an integer

The number of solid partitions of a positive integer is an unsolved problem in combinatorial number theory. In this paper, solid partitions are studied numerically by the method of exact enumeration for integers up to 50 and by Monte Carlo simulations using Wang-Landau sampling method for integers up to 8000. It is shown that, for large n, ln[p(n)]/n^(3/4) = 1.79 \pm 0.01, where p(n) is the number of solid partitions of the integer n. This result strongly suggests that the MacMahon conjecture for solid partitions, though not exact, could still give the correct leading asymptotic behaviour.

On minimal actions of Polish groups

We show the existence of an infinite monothetic Polish topological group G with the fixed point on compacta property. Such a group provides a positive answer to a question of Mitchell who asked whether such groups exist, and a negative answer to a problem of R. Ellis on the isomorphism of L( G), the universal point transitive G-system (for discrete G this is the same as βG the Stone-Čech compactification of G) and E( M, G), the enveloping semigroup of the universal minimal G-system ( M, G). For G with the fixed point on compacta property M is trivial while L( G) is not. Our next result is that even for Z with the discrete topology, L( Z) = βZ is not isomorphic to E( M, Z). Finally we show that the existence of a minimally almost periodic monothetic Polish topological group which does not have the fixed point property will provide a negative answer to an old problem in combinatorial number theory.

Some solved and unsolved problems in combinatorial number theory, ii

Some solved and unsolved problems in combinatorial number theory, ii

Lattice point simplices

We consider simplices in R m with lattice point vertices, no other boundary lattice points and n interior lattice points, with an emphasis on the barycentric coordinates of the interior points. We completely classify such triangles under unimodular equivalence and enumerate. For example, in a lattice point triangle with exactly one interior point, that point must be the centroid. We discuss the literature for fundamental tetrahedra and prove that there are seven possible barycentric coordinates for a one-point tetrahedron. Following suggestions of P. Erdös, we prove that, for fixed m and n, there are only finitely many possible sets of barycentric coordinates for the interior points. We also discuss a generalization of Beatty's problem in combinatorial number theory which has arisen several times in recent years.

Old and New Problems in Combinatorial Number Theory.

Old and New Problems in Combinatorial Number Theory.

Linear problems in combinatorial number theory

Linear problems in combinatorial number theory