Nonstandard Analysis for the Working Mathematician

Publisher Summary This chapter discusses the problems and results on combinatorial number theory. It discusses number theoretic problems that are of combinatorial nature. Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers along with the properties of objects made out of integers. Integers can be considered either in themselves or as solutions to equations. Questions in number theory are often understood through the study of analytical objects that encode properties of the integers, primes or other number-theoretic objects in some fashion. Real numbers can also be studied in relation to rational numbers.

In these notes we try to demonstrate the utility of the theory of theta functions in combinatorial number theory and complex analysis. The main idea is to use identities among theta functions to deduce either useful number-theoretic information related to representations as sums of squares and triangular numbers, statements concerning congruences, or statements concerning partitions of sets of integers. In complex analysis the main utility is in the theory of compact Riemann surfaces, with which we do not deal. We do show how identities among theta functions yield proofs of Picard’s theorem and a conformal map of the rectangle onto the disk.

The effective application of probabilistic reasoning in the study of problems in diverse areas is one of the most exciting recent developments in Mathematics. Probabilistic methods turned out to be very powerful in Discrete Mathematics, Analysis, Number Theory and Theoretical Computer Science. The meeting was dedicated to recent developments in these areas, focusing on the investigation of combinatorial problems for random sets and probabilistic methods, on the study of of questions in percolation, on the design and analysis of randomized algorithms and derandomization techniques, and on applications of probabilistic ideas in the study of questions in Combinatorial Number Theory and in Combinatorial Geometry.

Simultaneous systems of representatives and combinatorial number theory

New relationships between questions in combinatorial number theory are shown. The Erdos-Turan conjecture is shown to be equivalent to the finite problem; the Erdos-Turan conjecture for N is shown to follow from that for Z; and the additive number theory of the integers and that of the finite bit strings under bitwise mod 2 addition are contrasted. The dual space technique from coding theory is applied to the integer question, leading to a lower bound on the number of collisions. A subsidiary question in coding theory is raised, which is a refinement of the perfect code theorem. Finally some computations for cyclic codes are given.

A constructive approach to nonstandard analysis

Consider a codebook containing N unit-norm complex vectors in a K-dimensional space. In a number of applications, the codebook that minimizes the maximal cross-correlation amplitude (I/sub max/) is often desirable. Relying on tools from combinatorial number theory, we construct analytically optimal codebooks meeting, in certain cases, the Welch lower bound. When analytical constructions are not available, we develop an efficient numerical search method based on a generalized Lloyd algorithm, which leads to considerable improvement on the achieved I/sub max/ over existing alternatives. We also derive a composite lower bound on the minimum achievable I/sub max/ that is effective for any codebook size N.

A retrospective on 60 years of approximation theory and associated fields

We use ergodic-theoretical tools to study various notions of “large” sets of integers which naturally arise in theory of almost periodic functions, combinatorial number theory, and dynamics. Call a subset of N a Bohr set if it corresponds to an open subset in the Bohr compactification, and a piecewise Bohr set (PWB) if it contains arbitrarily large intervals of a fixed Bohr set. For example, we link the notion of PWB-sets to sets of the form A+B, where A and B are sets of integers having positive upper Banach density and obtain the following sharpening of a recent result of Renling Jin.

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.

Book Review: Recurrence in ergodic theory and combinatorial number theory

The main result of this paper is a coefficient formula that sharpens and generalizes Alon and Tarsi's Combinatorial Nullstellensatz. On its own, it is a result about polynomials, providing some information about the polynomial map $P|_{\mathfrak{X}_1\times\cdots\times\mathfrak{X}_n}$ when only incomplete information about the polynomial $P(X_1,\dots,X_n)$ is given.In a very general working frame, the grid points $x\in \mathfrak{X}_1\times\cdots\times\mathfrak{X}_n$ which do not vanish under an algebraic solution – a certain describing polynomial $P(X_1,\dots,X_n)$ – correspond to the explicit solutions of a problem. As a consequence of the coefficient formula, we prove that the existence of an algebraic solution is equivalent to the existence of a nontrivial solution to a problem. By a problem, we mean everything that "owns" both, a set ${\cal S}$, which may be called the set of solutions; and a subset ${\cal S}_{\rm triv}\subseteq{\cal S}$, the set of trivial solutions.We give several examples of how to find algebraic solutions, and how to apply our coefficient formula. These examples are mainly from graph theory and combinatorial number theory, but we also prove several versions of Chevalley and Warning's Theorem, including a generalization of Olson's Theorem, as examples and useful corollaries.We obtain a permanent formula by applying our coefficient formula to the matrix polynomial, which is a generalization of the graph polynomial. This formula is an integrative generalization and sharpening of:1. Ryser's permanent formula.2. Alon's Permanent Lemma.3. Alon and Tarsi's Theorem about orientations and colorings of graphs.Furthermore, in combination with the Vigneron-Ellingham-Goddyn property of planar $n$-regular graphs, the formula contains as very special cases:4. Scheim's formula for the number of edge $n$-colorings of such graphs.5. Ellingham and Goddyn's partial answer to the list coloring conjecture.

Let F q \mathbb {F}_q denote the finite field of characteristic p p and order q q . Let Z q \mathbb {Z}_q denote the unramified extension of the p p -adic rational integers Z p \mathbb {Z}_p with residue field F q \mathbb {F}_q . Given two positive integers m , n m,n , define a box B m \mathcal B_m to be a subset of Z q n \mathbb {Z}_q^n with q n m q^{nm} elements such that B m \mathcal B_m modulo p m p^m is equal to ( Z q / p m Z q ) n (\mathbb {Z}_q/p^m \mathbb {Z}_q)^n . For a collection of nonconstant polynomials f 1 , … , f s ∈ Z q [ x 1 , … , x n ] f_1,\dots ,f_s\in \mathbb {Z}_q[x_1,\ldots ,x_n] and positive integers m 1 , … , m s m_1,\dots ,m_s , define the set of common zeros inside the box B m \mathcal B_m to be V = { X ∈ B m : f i ( X ) ≡ 0 mod p m i for all 1 ≤ i ≤ s } . \begin{equation*} V=\{X\in \mathcal B_m:\; f_i(X)\equiv 0\mod {p^{m_i}}\text { for all } 1\leq i\leq s\}. \end{equation*} It is an interesting problem to give the sharp estimates for the p p -divisibility of | V | |V| . This problem has been partially solved for the three cases: (i) m = m 1 = ⋯ = m s = 1 m=m_1=\cdots =m_s=1 , which is just the Ax-Katz theorem, (ii) m = m 1 = ⋯ = m s > 1 m=m_1=\cdots =m_s>1 , which was solved by Katz [Proc. Amer. Math. Soc. 137 (2009), pp. 4065–4075; Amer. J. Math. 93 (1971), pp. 485–499], Marshal and Ramage [Proc. Amer. Math. Soc. 49 (1975), pp. 35–38], and (iii) m = 1 m=1 , and m 1 , … , m s ≥ 1 m_1,\dots ,m_s\geq 1 , which was recently solved by Cao, Wan [Finite Fields Appl. 91 (2023), p. 25] and Grynkiewicz [ A generalization of the Chevalley-Warning and Ax-Katz theorems with a view towards combinatorial number theory , Preprint, arXiv: 2208.12895 , 2022]. Based on the multi-fold addition and multiplication of the finite Witt rings over F q \mathbb {F}_q , we investigate the remaining unconsidered case of m > 1 m>1 and m ≠ m j m\neq m_j for some 1 ≤ j ≤ s 1\leq j\leq s , and finally provide a complete answer to this problem.

Ultrafilters, IP sets, dynamics, and combinatorial number theory

The investigation of inverse problems has a long tradition in combinatorial number theory (see [107, 37]), and more recently it has been promoted by applications in the theory of non-unique factorizations. In this chapter we discuss the inverse problems associated with the invariants D(G), η(G) and s(G). More precisely, we investigate the structure of sequences of length D(G)−1 (η(G)−1 or s(G)−1, respectively) that do not have a zero-sum subsequence (of the required length). Recent results on the structure of Σ(S) for (long) zero-sum free sequences may be found in [9, 58, 127, 132, 60].