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Über dieses Buch

This volume is dedicated to Harvey Cohn, Distinguished Professor Emeritus of Mathematics at City College (CUNY). Harvey was one of the organizers of the New York Number Theory Seminar, and was deeply involved in all aspects of the Seminar from its first meeting in January, 1982, until his retirement in December, 1995. We wish him good health and continued hapiness and success in mathematics. The papers in this volume are revised and expanded versions of lectures delivered in the New York Number Theory Seminar. The Seminar meets weekly at the Graduate School and University Center of the City University of New York (CUNY). In addition, some of the papers in this book were presented at a conference on Combinatorial Number Theory that the New York Number Theory Seminar organized at Lehman College (CUNY). Here is a short description of the papers in this volume. The paper of R. T. Bumby focuses on "elementary" fast algorithms in sums of two and four squares. The actual talk had been accompanied by dazzling computer demonstrations. The detailed review of H. Cohn describes the construction of modular equations as the basis of studies of modular forms in the one-dimensional and Hilbert cases.



1. Sums of Four Squares

The main results to be presented here deal with representations as a sum of four squares. However, it is useful for purposes of exposition to consider the corresponding theorems for sums of two squares. Since these results are so familiar, and part of elementary courses, it may seem that these propositions are belaboring the obvious. However, there is a slight difference in emphasis from the usual treatment that will be useful in describing the generalization. There are two types of questions to be considered: algorithmic — how can one compute representations of a number as a sum of two or four (or possibly some other number) of squares? — and enumerative — is there a structure on the set of representations that allows their number to be determined in an elementary manner?
Richard T. Bumby

2. On the Number of Co-Prime-Free Sets

For a variety of arithmetic properties P (such as the one in the title) we investigate the number of subsets of the positive integers ≤ x, that have that property. In so doing we answer some questions posed by Cameron and Erdös.
Neil J. Calkin, Andrew Granville

3. The Primary Role of Modular Equations

Recent developments indicate that the modular equation, historically the end product of modular function theory, should be considered as a starting point. For instance, while a Fourier series for a modular function has coefficients which determine a modular equation, conversely, modular functions are also determined by a modular equation. It is therefore of interest to ask what are the a priori determining features of a modular equation. Some features are revealed for the classic (one variable) case relating to normality of the parametrization. Other features are revealed for the Hilbert (two variable) case relating to the diagonals. A complete answer is still elusive, but a reasonable goal would be to find singular moduli directly (without modular invariants).
Harvey Cohn

4. Approximation Methods in Transcendental Function Computations and Some Physical Applications

High precision solution of extremal and complex analytic approximations problems that can be represented in terms of multiple integrals or integral equations involving hypergeometric functions are examined. Fast algorithms of computations of (approximate) solutions are presented that are well suited for parallelization. Among problems considered are: WKB and adelic asymptotics of multidimensional hypergeometric Padé approximations to classical functions, and high accuracy computations of high order eigenvalues and eigenstates for 2D and 3D domains of complex geometry. Methods based on boundary integrals, Galerkin techniques for various eigenfunction expansions and singularity analysis are examined.
D. V. Chudnovsky, G. V. Chudnovsky

5. Diophantine Approximation Problem Arising from VLSI Design

We want to focus attention on several related problems concerning deviation from the uniform distribution, arising in diophantine approximations, that are important for computer applications. These problems arise from the need to digitalize real-world data. In many such discretization problems diophantine approximation issues arise, especially when continuous data are described by quasiperiodic processes.
D. V. Chudnovsky, G. V. Chudnovsky

6. Linear Diophantine Problems

The Frobenius number g(A k ) Let A k \({A_k} = \{ {a_1},...,{a_k}\}\subset\) IN with gcd(A k ) = 1, n\( \in I{N_0}.\) If
$$n = \sum\limits_{i = 1}^k {{x_i}{a_i},{x_i}}\in I{N_0}$$
we call this a representation or a g-representation of n by Ak (in order to distinguish between several types of representations that will be considered in the sequel). Then the Frobenius number g(A k ) is the greatest integer with no g-representation.
Mehdi Djawadi, Gerd Hofmeister

7. On the Sum of the Reciprocals of the Differences Between Consecutive Primes

The infinite series
$$\sum\limits_{{n = 2}}^{\infty } {\frac{1}{{n{{{(\log \log n)}}^{c}}\log n}}}$$
converges if and only if c > 1. Let pn denote the n-th prime number. By the prime number theorem,
$$\sum\limits_{{i = 1}}^{n} {({{p}_{{i + 1}}} - {{p}_{i}}) = {{p}_{{n + 1}}} - 2 \sim n \log n,}$$
and so the difference between consecutive primes is on average logn. This suggests the question: For what values of c does the series
$$\sum\limits_{{n = 2}}^{\infty } {\frac{1}{{n{{{(\log \log n)}}^{c}}({{p}_{{n + 1}}} - {{p}_{n}})}}}$$
converge? We shall prove convergence for c > 2, and give a heuristic argument why the series must diverge for c = 2.
Paul Erdös, Melvyn B. Nathanson

8. The Smallest Maximal Set of Pairwise Disjoint Partitions

Let k be an integer ≥ 3, and let n be a natural number greater than k. Among the partitions of n into k positive integers, consider m = m k (n) partitions of n, say n = ai1 + ai2 + aik for i = l,2,…,m, satisfying the km integers of the set A = (aij: i = 1,…,m, j = 1,k) are distinct, if a1,ak. are any k distinct positive integers not in A, then a1 + ak-≠ n, and m =k(n) is minimal.
Michael Filaseta

9. Sum Set Cardinalities of Line Restricted Planar Sets

A theorem of G. Freiman says that if 2 ≤ λ < 4 then there isac=c(λ)>0 such that, for all finite sets X in the plane, if the sum set X+X has no more than λ|X| points then some line in the plane contains at least c|X| points of X. The present paper strengthens results used in Fishbum’s proof of Freiman’s theorem to obtain interestingly large values of c for which the theorem holds. For example, if the conclusion is to hold at least for suitably large |X|, then c=1 suffices for 2 ≤ X < 3, and any value of c < (4-λ)/2 suffices for 3 ≤ λ < 4. At the same time, if (4-λ)/2=l/k for k∈ {1,2,…}, then no value of c > 1/k can give the desired result, even when |X| is restricted to be large.
Peter C. Fishburn

10. On Solvability of a System of Two Boolean Linear Equations

The solvability of a system of two boolean linear equations is considered. The methods of analytical number theory allow us to characterize the set of right-hand sides for which the system has solutions. A new approach is applicable to systems whose coefficients are bounded relative to the number of unknowns and whose right-hand sides are in the certain wide neighborhood of the middle point of the sum of coefficients (unlike the dynamic programming which is well suited only for small right-hand sides). The new method can also be used to design efficient algorithms.
Gregory Freiman

11. “Brauer Numbers” of Twisted Fermat Motives

Without Abstract
Fernando Q. Gouvea, Noriko Yui

12. A Remark on a Paper of Erdős and Nathanson

A set A of integers is said to be an asymptotic basis of order h if every sufficiently large integer can be represented as a sum of h (not necessarily instinct) elements of A. In a recent paper [ENJ, Erdös and Nathanson prove the following interesting result.
R. L. Graham

13. Towards a Classification of Hilbert Modular Threefolds

We begin with the classical (full) modular group and variety of which Hilbert modular groups and varieties are generalizations.
H. G. Grundman

14. Special Theta Relations

§My topic arises out of my efforts to respond to a question raised last summer by W. L. Hoyt at the AMS Summer Institute on Theta Functions: How can theta nullwerte be used to describe Hilbert modular surfaces, or, more generally, Humbert surfaces, as subvarieties of the variety of moduli of abelian surfaces (e. g., with principal polarization)? I want to show that an easy very slight re-packaging of the presentation by Mumford [4] of the subject of Riemann’s relations, which are general quartic relations satisfied by the homogeneous coordinates (for suitable projective embeddings) of the moduli of all abelian varieties, leads to special quadratic relations for these homogeneous coordinates when the moduli represent abelian varieties that are special in a certain sense.
William F. Hammond

15. Minimal Bases and g-Adic Representations of Integers

Let A be a set of integers, h ≥ 2 an integer. Let hA denote the set of all sums of h elements of A. If hA contains all sufficiently large integers, then A is called an asymptotic basis of order h. An asymptotic basis A of order h is said to be minimal if it contains no proper subset which is again an asymptotic basis of order h. This concept of minimality of bases was first introduced by Stöhr [5]. Härtter [1] showed the existence of minimal asymptotic bases by a nonconstructive argument. Nathanson [3] constructed the first nontrivial example of minimal asymptotic bases of order h ≥ 2. Jia and Nathanson [2] recently discovered a simple construction of minimal asymptotic bases of order h ≥ 2 by using powers of 2. Furthermore, for any α: 1/h ≤; α < 1, they constructed a minimal asymptotic basis A of order h such that x α < A(x) < x α, where A(x) is the number of positive elements not exceeding x. In the present paper, we shall generalize these results to g-adic representations of integers.
Xing-De Jia

16. Finite Graphs and the Number of Sums and Products

Let G be a graph with k vertices (1, 2, …, k) and e edges. Let A = (α12,..,α k ) be a set of k integers, and let G(A) be the set of all integers of the form α i + α j and α i α j , where (i,j) is an edge of G. Erdös and Szemerédi conjectured that |G(α)| ≫ ε e /k ε for every ε > 0 and every set A. This conjecture will be proved in the case that the diameter of the set A is polynomial in k.
Xing-De Jia, Melvyn B. Nathanson

17. Hilberth’s Theorem 94 and Function Fields

Let f(x,y) be a monic absolutely irreducible polynomial in x of degree n with coefficients in Z[y]. If α is a root of f(x,y), L = Q(y)(α)/Q(y). Hilbert’s Theorem 94 [4] gives a procedure for determining rational primes p which divide the class number of a number field. Here an analogue of it is given for ordinary arithmetic function fields like L as defined by E. Weiss in [7]. A corollary of Theorem 1 is used to obtain rational prime divisors of class numbers of number fields L’ obtained from L by specialization of y into Z. Although the proof essentially follows that of Hilbert, use is made of the concept of NTU#x2019;s (non-trivial units) in fields like L. These units were implicity defined in [5].
Howard Kleiman

18. Some Applications of Probability to Additive Number Theory and Harmonic Analysis

We present some applications of the probabilistic method in additive number theory and harmonic analysis. We describe three general approaches to the probablistic construction of certain objects. The question of whether one can actually “construct” these is also discussed and several examples of “derandomized” probabilistic proofs are given.
Mihail N. Kolountzakis

19. Quadratic Irrationals and Continued Fractions

Quadratic irrationals with purely periodic continued fractions \(\overline {\left[ {{b_0},{b_1}, \ldots ,{b_{k - 1}}} \right],} \) where b 0, …, b k-1, or b 1, …, b k-1 is a palindrome are studied. For each d the number of such irrationals having discriminant d is determined and the relation between these two types is given.
Joseph Lewittes

20. Progression Bases for Finite Cyclic Groups

A set A of integers is an additive basis modulo n if every integer is congruent mod n to a sum of at most h elements of A, repetitions being allowed. The set A is a basis of order h in case h is minimal. In this paper we study the order of bases of the form (α, 2α, …, ) U (b, 2b, …, 1b) and of the form (α, α + b, α + 2b, …, α + kb), where α, b are integers satisfying gcd(α, b, n) = 1.
Öystein J. Rödseth

21. Sums of Finite Sets

We investigate numerous cardinality questions concerning sums of finite sets. A typical problem looks like the following: if A has n elements, A + B has cn, what can we deduce about A and B? How can we estimate the cardinalities of other sets like AB and A + B + A? This is in quest of a generalization of Freiman’s famous theorem that describes the structure of those sets A for which A + A is small, to the case of different summands.
Imre Z. Ruzsa

22. Four Squares with Few Squares

The classic theorem of Lagrange states that every nonnegative integer n is the sum of four squares. How “sparse” can a set of squares be and still retain the four square property. For any set X of nonnegative integers set Nx(x) = |{i ∈ X, ≤x}|. Let S = {0,1,4,9,…} denote the squares. If \(X \subseteq S\) and every n ≥ 0 can be expressed as the sum of four elements of X then how slow can be the growth rate of Nx(x) ? Clearly we must have = N(x1/4). Our object here is to give a quick proof of the following result of Wirsing[3]
Joel Spencer
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