Number Theory: New York Seminar 1991–1995

291 Pages·1996·6.268 MB·English

by  Richard T. Bumby (auth.), David V. Chudnovsky, Gregory V. Chudnovsky, Melvyn B. Nathanson (eds.)

Number Theory
New York Seminar 1991-1995
Springer
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Berlin
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Budapest
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London
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Singapore
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D.V. Chudnovsky G.V. Chudnovsky
M.B. Nathanson
Editors
Number Theory
New York Seminar 1991-1995
With 78 Figures
, Springer
David V. Chudnovsky Melvyn B. Nathanson
Gregory V. Chudnovsky Department of Mathematics
Department of Mathematics Lehman College
Columbia University City University of New York
New York, NY 10027 Bronx, NY 10468
USA USA
Mathematics Subject Classifications (1991): lO-06
Library of Congress Cataloging-in-Publication Data
Number theory: New York seminar, 1991-1995/ David V. Chudnovsky,
Gregory V. Chudnovsky, Melvyn B. Nathanson [editors}.
p. cm.
Includes bibliographical references.
ISBN 9784),387-94826-3 (soft: alk. paper)
1. Number theory-Congresses. I. Chudnovsky, D. (David), 1947-
II. Chudnovsky, G. (Gregory), 1952- . III. Nathanson, Melvyn
B. (Melvyn Bernard), 1944-
QA241.N8743 1996
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[SBN-[3: 978-0-387-94826-3 e-[SBN: 978-[-46[2-2418-1
001: 10.1007/978-1-4612-2418-1
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ISBN 978-D-387-94826-3 Springer-Verlag New York Berlin Heidelberg SPIN 10523547
To Harvey Cohn
Preface
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.
The first paper of D. V. and G. V. Chudnovsky deals with fast approximation
techniques for special functions needed in eigenvalue and eigenfunction analysis
of different domains, with applications to number theory. The second paper
of these authors applies diophantine approximations to build various sequences
in 1D and 2D with small discrepancies. These sequences, behaving like "finite
quasi-crystals," are applied to computer problems from VLSI designs.
Many authors have investigated the gaps between consecutive primes, that
is, the numbers Pn+l - Pn. P. Erdos and M. B. Nathanson prove that the series
En l/{n{loglogn)c(pn+l - Pn)) converges for c > 2, and they indicate why this
series almost certainly diverges for c = 2. A paper by F. Q. Gouvea and N. Yui
computes explicitly the "Brauer number" and the order of the Brauer group for
Fermat varieties. H. G. Grundman classifies Hilbert modular threefolds over
cubic fields. W. F. Hammond's paper presents quadratic relations among theta
nullwerte characterizing Hilbert modular surfaces. The paper of H. Kleiman
uses Hilbert's Theorem 94 to find non-trivial units in function fields. This is
needed in diophantine geometry. J. Lewittes classifies quadratic irrationalities
that have pure palindromic periods in their continued fraction expansions.
Combinatorial number theory is the subject of many papers in this volume.
A general problem is to estimate the number of sets of positive integers not
viii Preface
exceeding x that have some property. N. J. Calkin and A. Granville investigate
this in several cases, for example, when the property is that no two elements in
the set are relatively prime. M. Djawadi and G. Hofmeister's paper is connected
with the following well-known problem of Frobenius: Let aI, a2, ... ,ak be rela
tively prime positive integers. What is the largest integer g( aI, a2, ... , ak) that
cannot be written as a nonnegative integral linear combination of the ai's?
M. Filaseta constructs small maximal sets of pairwise disjoint partitions of a
positive integer into k distinct parts, and improves a lower bound of Nathanson.
P. C. Fishburn considers a problem in combinatorial geometry that arises in
the study of inverse problems in additive number theory. G. Freiman applies
methods from analytic number theory to study the solvability of Boolean linear
equations. R. L. Graham investigates a combinatorial problem that arises in
work of Erdos and Nathanson on partitioning asymptotic bases into pairwise
disjoint asymptotic bases. X. Jia uses g-adic representations to construct a new
class of minimal asymptotic bases of order h.
Erdos and Szemeredi introduced the problem of estimating the number of
integers that can be written as either the pairwise sum or the pairwise product
of elements in a fixed set of k positive integers. They conjectured that this
»
number is k2-e:. X. Jia and M. B. Nathanson investigate a graph-theoretic
version of this conjecture. M. N. Kolountzakis presents a survey of applications
of probability methods to additive number theory and harmonic analysis, with
particular emphasis on the effort to "derandomize" probabilistic arguments in
an attempt to make them constructive. O. J. R6dseth examines minimal bases
in cyclic groups.
I. Z. Ruzsa's paper investigates various possible generalizations of Freiman's
famous inverse theorem in additive number theory, and applications of a graph
theoretic method of Pliinnecke to these problems. Finally, J. Spencer applies
a recently discovered correlation inequality to give a short and elegant proof of
the existence of thin subsequences of squares that are bases of order four for the
natural numbers.
This is the sixth in a series of books that have come out of the New York
Number Theory Seminar. We wish to thank Springer-Verlag for its continuing
interest in this project.
The Seminar has been supported in part by a grant from the Mathematical
Sciences Program of the National Security Agency. We are grateful to NSA for
its support.
Contents
Preface ............................................................. vii
1 Sums of Four Squares ................................................ 1
R. T. Bumby
2 On the Number of Co-Prime-Free Sets ................................ 9
N. J. Calkin and A. Granville
3 The Primary Role of Modular Equations ............................ 19
H. Cohn
4 Approximation Methods in 'Transcendental Function Computations
and Some Physical Applications ..................................... 43
D. V. Chudnovsky and G. V. Chudnovsky
5 Diophantine Approximation Problem Arising From VLSI Design ..... 71
D. V. Chudnovsky and G. V. Chudnovsky
6 Linear Diophantine Problems ........................................ 91
M. Djawadi and G. Hofmeister
7 On the Sum of the Reciprocals of the Differences Between
Consecutive Primes ................................................. 97
P. Erdos and M. B. Nathanson
8 The Smallest Maximal Set of Pairwise Disjoint Partitions ........... 103
M. Filaseta
9 Sum Set Cardinalities of Line Restricted Planar Sets ............... 115
P. C. Fishburn
10 On Solvability of a System of Two Boolean Linear Equations ....... 135
G . .Freiman
11 Brauer Number and Twisted Fermat Motives ....................... 151
F. Q. Gouvea and N. Yui
12 A Remark on a Paper of Erdos and Nathanson ..................... 177
R. L. Graham
13 Towards a Classification of Hilbert Modular Threefolds ............. 181
H. G. Grundman
14 Special Theta Relations ............................................ 195
W. F. Hammond
x Contents
15 Minimal Bases and g-adic Representations of Integers .............. 201
X-D. Jia
16 Finite Graphs and the Number of Sums and Products .............. 211
X-D. Jia and M. B. Nathanson
17 Hilbert's Theorem 94 and Function Fields .......................... 221
H. Kleiman
18 Some Applications of Probability to Additive Number Theory
and Harmonic Analysis ............................................ 229
M. N. K olountzakis
19 Quadratic Irrationals and Continued Fractions ..................... 253
J. Lewittes
20 Progression Bases for Finite Cyclic Groups ......................... 269
O. J. Rodseth
21 Sums of Finite Sets ................................................ 281
1. Z. Ruzsa
22 Four Squares with Few Squares .................................... 295
J. Spencer
1
Sums of Four Squares
Richard T. Bumby'
O. Introduction
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?
The algorithmic question is treated to a certain extent in elementary texts. At this
level, only the question of representing primes is considered and the question of the speed
of the algorithm is generally ignored. Nonetheless, the usual algorithm for sums of two
squares is polynomial-time relative to finding a square root of -1 modulo the number
to be represented. For sums of four squares, the situation is a little different. Some
books give algorithms which, while similar to that for sums of two squares, fail to be
polynomial-time. Other books modify the algorithm so that it becomes polynomial-time
if one has found an expression of -1 as a sum of two squares modulo the number to be
represented. Strangely, no comment seems to be made on this distinction although the
speed of algorithms is generally considered an important problem at the present time.
Furthermore, the emphasis in textbooks is entirely on the representation of primes. This
goes back to the early work on the subject. The formula for producing representations
of products from representations of the factors appears to reduce the question to that of
representing primes. However, questions of computational complexity make it clear that
there may be some benefits in avoiding factorization.
The development to be described here suggests that the proper role of factorization
in the question of representation as a sum of four squares occurs at the level of finding
a representation of -1 as a sum of two squares modulo the number to be represented.
Recently, I found an fast, sure and elementary procedure for fining such a representation
modulo a prime (in many different ways). Thus, representing a number as a sum of four
squares is no more difficult than factoring the number. It would be interesting to know its
true complexity.
In some sense, the simpler problem of representations as a sum of two squares is
more complicated. There are fast probabilistic algorithms for finding square roots modulo
primes, which allows fast, but not sure, elementary computations of primes as a sum of
two squares. In addition, the algorithm of Schoof [6] gives a deterministic polynomial time
computation of a prime congruent to 1 mod 4 as a sum of two squares. This method is
fast and sure, but I would not consider it to be elementary.
The enumerative question has been solved using modular forms. This allows exact
formulas to be found for the number of representations of a number as a sum of 2, 4, 6 or
I Rutgers University.