Ebook: Irregularities of Partitions

The problem of uniform distribution of sequences initiated by Hardy, Little­ wood and Weyl in the 1910's has now become an important part of number theory. This is also true, in relation to combinatorics, of what is called Ramsey­ theory, a theory of about the same age going back to Schur. Both concern the distribution of sequences of elements in certain collection of subsets. But it was not known until quite recently that the two are closely interweaving bear­ ing fruits for both. At the same time other fields of mathematics, such as ergodic theory, geometry, information theory, algorithm theory etc. have also joined in. (See the survey articles: V. T. S6s: Irregularities of partitions, Lec­ ture Notes Series 82, London Math. Soc. , Surveys in Combinatorics, 1983, or J. Beck: Irregularities of distributions and combinatorics, Lecture Notes Series 103, London Math. Soc. , Surveys in Combinatorics, 1985. ) The meeting held at Fertod, Hungary from the 7th to 11th of July, 1986 was to emphasize this development by bringing together a few people working on different aspects of this circle of problems. Although combinatorics formed the biggest contingent (see papers 2, 3, 6, 7, 13) some number theoretic and analytic aspects (see papers 4, 10, 11, 14) generalization of both (5, 8, 9, 12) as well as irregularities of distribution in the geometric theory of numbers (1), the most important instrument in bringing about the above combination of ideas are also represented.

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The problem of the uniform distribution of sequences, first attacked by Hardy, Littlewood and Weyl in the early years of this century, has now become an important part of number theory. This is also true of Ramsey theory in combinatorics, whose origins can be traced back to Schur in the same period. Both concern the distribution of sequences of elements in certain collection of subsets. Quite recently these strands have become interwoven, borne fruit and developed links with such other fields as ergodic theory, geometry, information theory and algorithm theory. This volume is the homogeneous summary of a workshop held at Fert?d in Hungary, which brought together people working on various aspects of Ramsey theory on the one hand and on the theory of uniform distribution and related aspects of number theory on the other. The volume consists of 14 papers, 5 on the combinatorial, 5 on the number theoretical aspects and 4 on various generalizations, and a list of unsolved problems. This authoritative state-of-the-art report is addressed to researchers and graduate students.

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The problem of the uniform distribution of sequences, first attacked by Hardy, Littlewood and Weyl in the early years of this century, has now become an important part of number theory. This is also true of Ramsey theory in combinatorics, whose origins can be traced back to Schur in the same period. Both concern the distribution of sequences of elements in certain collection of subsets. Quite recently these strands have become interwoven, borne fruit and developed links with such other fields as ergodic theory, geometry, information theory and algorithm theory. This volume is the homogeneous summary of a workshop held at Fert?d in Hungary, which brought together people working on various aspects of Ramsey theory on the one hand and on the theory of uniform distribution and related aspects of number theory on the other. The volume consists of 14 papers, 5 on the combinatorial, 5 on the number theoretical aspects and 4 on various generalizations, and a list of unsolved problems. This authoritative state-of-the-art report is addressed to researchers and graduate students.
Content:
Front Matter....Pages i-vii
Irregularities of Point Distribution Relative to Convex Polygons....Pages 1-22
Balancing Matrices with Line Shifts II....Pages 23-37
A Few Remarks on Orientation of Graphs and Ramsey Theory....Pages 39-46
On a Conjecture of Roth and Some Related Problems I....Pages 47-59
Discrepancy of Sequences in Discrete Spaces....Pages 61-70
On the Distribution of Monochromatic Configurations....Pages 71-87
Covering Complete Graphs by Monochromatic Paths....Pages 89-91
Canonical Partition Behaviour of Cantor Spaces....Pages 93-105
Extremal Problems for Discrepancy....Pages 107-113
Spectral Studies of Automata....Pages 115-128
A Diophantine Problem....Pages 129-135
A Note on Boolean Dimension of Posets....Pages 137-140
Intersection Properties and Extremal Problems for Set Systems....Pages 141-151
On an Imbalance Problem in the Theory of Point Distribution....Pages 153-160
Problems....Pages 161-165
Back Matter....Pages 167-168

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The problem of the uniform distribution of sequences, first attacked by Hardy, Littlewood and Weyl in the early years of this century, has now become an important part of number theory. This is also true of Ramsey theory in combinatorics, whose origins can be traced back to Schur in the same period. Both concern the distribution of sequences of elements in certain collection of subsets. Quite recently these strands have become interwoven, borne fruit and developed links with such other fields as ergodic theory, geometry, information theory and algorithm theory. This volume is the homogeneous summary of a workshop held at Fert?d in Hungary, which brought together people working on various aspects of Ramsey theory on the one hand and on the theory of uniform distribution and related aspects of number theory on the other. The volume consists of 14 papers, 5 on the combinatorial, 5 on the number theoretical aspects and 4 on various generalizations, and a list of unsolved problems. This authoritative state-of-the-art report is addressed to researchers and graduate students.
Content:
Front Matter....Pages i-vii
Irregularities of Point Distribution Relative to Convex Polygons....Pages 1-22
Balancing Matrices with Line Shifts II....Pages 23-37
A Few Remarks on Orientation of Graphs and Ramsey Theory....Pages 39-46
On a Conjecture of Roth and Some Related Problems I....Pages 47-59
Discrepancy of Sequences in Discrete Spaces....Pages 61-70
On the Distribution of Monochromatic Configurations....Pages 71-87
Covering Complete Graphs by Monochromatic Paths....Pages 89-91
Canonical Partition Behaviour of Cantor Spaces....Pages 93-105
Extremal Problems for Discrepancy....Pages 107-113
Spectral Studies of Automata....Pages 115-128
A Diophantine Problem....Pages 129-135
A Note on Boolean Dimension of Posets....Pages 137-140
Intersection Properties and Extremal Problems for Set Systems....Pages 141-151
On an Imbalance Problem in the Theory of Point Distribution....Pages 153-160
Problems....Pages 161-165
Back Matter....Pages 167-168
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