Problems and results on combinatorial number theory III

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.

Starting with a simple formulation accessible to all mathematicians, this second edition is designed to provide a thorough introduction to nonstandard analysis. Nonstandard analysis is now a well-developed, powerful instrument for solving open problems in almost all disciplines of mathematics; it is often used as a secret weapon by those who know the technique. This book illuminates the subject with some of the most striking applications in analysis, topology, functional analysis, probability and stochastic analysis, as well as applications in economics and combinatorial number theory. The first chapter is designed to facilitate the beginner in learning this technique by starting with calculus and basic real analysis. The second chapter provides the reader with the most important tools of nonstandard analysis: the transfer principle, Keisler s internal definition principle, the spill-over principle, and saturation. The remaining chapters of the book study different fields for applications; each begins with a gentle introduction before then exploring solutions to open problems. All chapters within this second edition have been reworked and updated, with several completely new chapters on compactifications and number theory. Nonstandard Analysis for the Working Mathematician will be accessible to both experts and non-experts, and will ultimately provide many new and helpful insights into the enterprise of mathematics

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.

Dear Reader: The Moscow Journal of Combinatorics and Number Theory was founded in 2010 by the Moscow Institute of Physics and Technology, and in 2018 it started being published by MSP (Mathematical Sciences Publishers), a nonprofit scientific publisher based in Berkeley, California. Our journal publishes original, high-quality research articles from a broad range of interests within combinatorics, number theory and allied areas. Since 2011 we have published over 100 papers. Among our authors are such mathematicians as Noga Alon, Antal Balog, Jean-Pierre Demailly, Dominic Foata, Peter Frankl, Aleksandar Ivic, Sergei Konyagin, Yuri Nesterenko, Janos Pach, Yakov Sinai, Andrzej Schinzel, Wolfgang Schmidt, Carlo Viola, Michel Waldschmidt, and many others. This issue 1 of volume 8 is the first issue to appear under MSP's aegis. It contains selected papers presented by the participants of the Vilnius Conference in Combinatorics and Number Theory, which was organized with support from our journal and took place at the Department of Mathematics and Informatics of the University of Vilnius, Lithuania, 16–22 July 2017. Previous conferences connected to our journal were held in Russia (Diophantine analysis, Astrakhan, 30 July to 3 August 2012), Lithuania (Palanga Conference in Combinatorics and Number Theory, 1–7 September 2013), again Russia (Moscow Workshop in Combinatorics and Number Theory, January 27 to 2 February 2014), and Denmark (Diophantine Approximation and Related Topics, Aarhus, 13–17 July 2015). A collection of papers from the Astrakhan conference appeared in issue 3–4 of volume 3 (2013), and papers related to the Aarhus conference in issue 2–3 of volume 6 (2016). We hope that you will enjoy this issue and support the journal both with your submissions and by recommending a subscription to your institutional library!

The Thomist 64 (2000): 101-25 CANTOR'S TRANSFINITE NUMBERS AND TRADITIONAL OBJECTIONS TO ACTUAL INFINITY JEANW. RIOUX Benedictine College Atchison, Kansas Georg Cantor was one of the most prominent mathematicians of the late nineteenth and early twentieth centuries. His development of a theory of transfinite numbers resurrected philosophical questions about infinity and led to a division of mathematics into schools of thought such as formalism and intuitionism. Cantor's published attempts to justify his mathematical theories were directed not only toward the mathematicians of his day but also toward philosophers, both ancient and contemporary. His efforts on the latter front were rooted in his desire to deal with objections to the very idea of the actual infinite in quantity, and he attached great importance to those objections that came from traditional philosophy.1 I intend to review the basic philosophical issues: Cantor's claims to a workable mathematics of real and actually infinite quantities, his response to Aristotelian objections to those claims, and my reflections on whether Cantor finally settled the matter, as he had hoped. 1 I shall focus upon Cantor's philosophical defense of transfinite numbers against this fundamental Aristotelian position as it is found in part 5 of his "Ueber unendliche, lineare Punktmannigfaltigkeiten," Mathematische Annalen 21 :545-86 (published separately as Grundlagen einer allgemeinen Mannigfaltigkeitslehre. Ei11 mathematisch-philosophischer Versuch in der Lehre des Unend.lichen [Leipzig: B. G. Teubner, 1883], hereafter Grumilage11). Cantor saw Aristotle as the source of the Scholastic position on infinity, and in the Gnmdlagen he addressed the basic error involved in all 'finitist' reasoning, as he saw it. A few years after writing the Grundlagen, Cantor published papers dealing with specific Scholastic arguments against the actual infinite in detail. For example, see Robin Small, "Cantor and the Scholastics," American Catholic Philosophical Quarterly 66 (1992): 407-28, where the question concerns the eternity of the world. 101 102 JEANW. RIOUX Cantor was well aware of the prevailing mathematical and philosophical climates of his day.2 As he drew nearer to a completed theory of transfinite numbers, he became increasingly interested in justifying it, not only as a consistent and practical3 exercise of mathematical thought, but also as one having a basis in the real world,4 as somehow providing as real a view of the natural world as did the relatively unproblematic theory of finite integers. While other mathematicians seemed unconcerned with such metaphysical questions, Cantor devoted much of his time and effort to addressing them, especially later in his life.5 Ironically, Cantor'sfascination with these metaphysical aspects of the theory, so foreign to his contemporaries in mathematics, turned out to be somewhat prophetic, since the dubious ontological character of transfinite numbers was to figure prominently in later developments in mathematics.6 I will begin by giving an overview of Cantor's transfinite number theory, focusing in particular upon his claim that transfinite numbers possess an objective, or real, infinity which is actual, not merely potential.7 I. SUMMARY OF CANTOR'S TRANSFINITE NUMBER THEORY Cantor discovered thatmerely 'potential' infinity, the sortwith which mathematicians were comfortable to that day, was not the 2 See Joseph Warren Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite (Princeton, N.J.: Princeton University Press, 1979), 120-22. 3 As he notes in Grundlagen, § 8, the success in application of a mathematical concept is a major factor in whether it is accepted as legitimate or abandoned. 4 Cantor meant atleastthree thingsby 'real number': 'real' as distinguished from complex, rational, or irrational numbers; 'real' as existing in the understanding, as consistent and definite ideas in the mind; and 'real' as existing in the extramental world. Cantor held that all numbers that are real in the second sense are images of those that are real in the third sense. For more on the distinction between the last two types of reality, see Grundlagen, § 8. See also Dauben, Georg Cantor, 125-26. 5 Though many of the mathematicians of his day were able to look past Cantor's concern with the philosophical and theological implications of his transfinite theory (and in this way many came to defend the theory), to Cantor they were an inseparable part of his...

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