Monday, 24 February 2020
We start with the pigeonhole principle, of which the classical Ramsey theorem is a generalization. We then describe the philosophy of Ramsey Theory and give a proof of Ramsey's theorem.
In this lecture, we introduce the basic notions and notation regarding Sumsets and Inverse Additive Theory. The main focus will then be Kneser's Theorem, the fundamental classical result describing the structure of sumsets with very small sumset, followed by an exemplary application and a brief overview of some of it's many generalizations.
We begin with examples of typical questions in sieve theory (such as twin primes and Goldbach problem). These will lead to a general formulation of the main tasks in sieve theory. Then we study the Selberg upper bound sieve, whose basic idea is amazingly simple (as compared to Brun's sieve, for example). We apply it to particular sieve problems (such as the ones mentioned above). At the end of this lecture series we describe how Selberg's method can be turned into a lower bound sieve.
Literature: "An Introduction to Sieve Methods and their Applications" by A.C. Cojucaru and R. Murty, "Opera de Cribro" by J. Friedlander and H. Iwaniec, "Sieve Methods" by H. Halberstam and H.E. Richert.
The purpose of this series of lectures is to give an overview of zero-sum problems starting from classical results to questions of current research. A basic problem in zero-sum theory is the following problem: given a finite abelian group (G, +, 0) what is the smallest l such that each sequence of l elements over G has a non-empty subsequence whose terms sum to 0. This l is called the Davenport constant of G, denoted D(G). A classical variant of this problem is obtained by imposing a condition on the length of the subsequences: given a finite abelian group (G, +, 0) what is the smallest l such that each sequence of l elements over G has a subsequence of length exp(G) whose terms sum to 0. This l is called the Erd ̋os–Ginzburg–Ziv constant of G, denoted s(G).
The value of both constants is known for groups of rank at most two, yet not for general groups of rank three or higher.
A first goal is to present the proofs for groups of rank at most two. There are two main parts. First, arguments based on counting constellations of zero-sum subsequences, which typically are obtained via group-ring or polynomial methods. These yield result for p-groups. Second, inductive arguments, which allow to obtain insights on zero-sum constants for a group G from knowledge of zero-sum constants for a subgroup H and the quotient group G/H. A second goal is to review further results to obtain a good overview of what
is currently known on these constants.
Additional subjects to be discussed are the analogous problems for sets in-stead of sequences, the analogous problems for weighted sequences, and (time permitting) some applications.
Tentative schedule and selected references:
1. Introduction and overview
2. Davenport constant for p-groups, related results (towards a proof of Kemnitz conjecture)
3. Proof of Kemnitz conjecture and inductive arguments
4. Some recent advances on zero-sum constants
5. Problems with weights
6. Set-based problems (Olson constant, Harborth constant)
7. Applications, summary, perspectives.
Selected references:
1. S. D. Adhikari, Plus-minus weighted zero-sum constans: a survey. In: Analytic Number Theory, Modular Forms and q-Hypergeometric Series (G. E. Andrews and F. Garvan, eds), Springer 2018.
2. W. D. Gao and A. Geroldinger, Zero-sum problems in finite abelian groups: a survey. Expo. Math. 24 (2006), 337–369.
3. A. Geroldinger, Additive group theory and non-unique factorizations. In: Combinatorial Number Theory and Additive Group Theory (A. Geroldinger and I. Z. Ruzsa, eds), Birkh ̈auser 2009.
Tuesday, 25 February 2020
We take up van der Waerden's theorem and sketch a proof. We then state the Hales-Jewett theorem and discuss some variations/strengthenings of van der Waerden's theorem.
Kneser's Theorem is an extremely useful structural description of sumsets with $|A+B|<|A|+|B|-1$. In this lecture, we continue with the more challenging and detailed theory completely characterizing sumsets with $|A+B|\leq |A|+|B|$. We will also discuss some of the key ideas utilized in the proof of the characterization, including dual pairs (or trios) and additive energy.
We begin with examples of typical questions in sieve theory (such as twin primes and Goldbach problem). These will lead to a general formulation of the main tasks in sieve theory. Then we study the Selberg upper bound sieve, whose basic idea is amazingly simple (as compared to Brun's sieve, for example). We apply it to particular sieve problems (such as the ones mentioned above). At the end of this lecture series we describe how Selberg's method can be turned into a lower bound sieve.
Literature: "An Introduction to Sieve Methods and their Applications" by A.C. Cojucaru and R. Murty, "Opera de Cribro" by J. Friedlander and H. Iwaniec, "Sieve Methods" by H. Halberstam and H.E. Richert.
The purpose of this series of lectures is to give an overview of zero-sum problems starting from classical results to questions of current research. A basic problem in zero-sum theory is the following problem: given a finite abelian group (G, +, 0) what is the smallest l such that each sequence of l elements over G has a non-empty subsequence whose terms sum to 0. This l is called the Davenport constant of G, denoted D(G). A classical variant of this problem is obtained by imposing a condition on the length of the subsequences:
given a finite abelian group (G, +, 0) what is the smallest l such that each sequence of l elements over G has a subsequence of length exp(G) whose terms sum to 0. This l is called the Erd ̋os–Ginzburg–Ziv constant of G, denoted s(G). The value of both constants is known for groups of rank at most two, yet not for general groups of rank three or higher.
A first goal is to present the proofs for groups of rank at most two. There are two main parts. First, arguments based on counting constellations of zero sum subsequences, which typically are obtained via group-ring or polynomial methods. These yield result for p-groups. Second, inductive arguments, which allow to obtain insights on zero-sum constants for a group G from knowledge of zero-sum constants for a subgroup H and the quotient group G/H. A second goal is to review further results to obtain a good overview of what
is currently known on these constants.
Additional subjects to be discussed are the analogous problems for sets in-stead of sequences, the analogous problems for weighted sequences, and (time permitting) some applications. Tentative schedule and selected references:
1. Introduction and overview
2. Davenport constant for p-groups, related results (towards a proof of Kemnitz conjecture)
3. Proof of Kemnitz conjecture and inductive arguments
4. Some recent advances on zero-sum constants
5. Problems with weights
6. Set-based problems (Olson constant, Harborth constant)
7. Applications, summary, perspectives.
Selected references:
1. S. D. Adhikari, Plus-minus weighted zero-sum constans: a survey. In: Analytic Number Theory, Modular Forms and q-Hypergeometric Series (G. E. Andrews and F. Garvan, eds), Springer 2018.
2. W. D. Gao and A. Geroldinger, Zero-sum problems in finite abelian groups: a survey. Expo. Math. 24 (2006), 337–369.
3. A. Geroldinger, Additive group theory and non-unique factorizations. In:Combinatorial Number Theory and Additive Group Theory (A. Geroldinger and I. Z. Ruzsa, eds), Birkh ̈auser 2009.
Wednesday, 26 February 2020
We take up a theorem of Hilbert, perhaps the earliest Ramsey-type theorem, and give a topological proof.
The original version of Freiman's Theorem approximates the structure of integer sumsets $A+A$ when $|A+A|$ has bounded linear size. When $|A+A|\leq 3|A|-4$, more precise approximation results are available. This result is known as the Freiman $3k-4$ Theorem. In this lecture, we will talk about this result and its many generalizations, including versions for distinct summands, multiple summands, extensions with weaker hypotheses, and extensions to torsion groups via Fourier analytic methods.
We begin with examples of typical questions in sieve theory (such as twin primes and Goldbach problem). These will lead to a general formulation of the main tasks in sieve theory. Then we study the Selberg upper bound sieve, whose basic idea is amazingly simple (as compared to Brun's sieve, for example). We apply it to particular sieve problems (such as the ones mentioned above). At the end of this lecture series we describe how Selberg's method can be turned into a lower bound sieve.
Literature: "An Introduction to Sieve Methods and their Applications" by A.C. Cojucaru and R. Murty, "Opera de Cribro" by J. Friedlander and H. Iwaniec, "Sieve Methods" by H. Halberstam and H.E. Richert.
The purpose of this series of lectures is to give an overview of zero-sum problems starting from classical results to questions of current research. A basic problem in zero-sum theory is the following problem: given a finite abelian group (G, +, 0) what is the smallest l such that each sequence of l elements over G has a non-empty subsequence whose terms sum to 0. This l is called the Davenport constant of G, denoted D(G). A classical variant of this problem is obtained by imposing a condition on the length of the subsequences:
given a finite abelian group (G, +, 0) what is the smallest l such that each sequence of l elements over G has a subsequence of length exp(G) whose terms sum to 0. This l is called the Erd ̋os–Ginzburg–Ziv constant of G, denoted s(G). The value of both constants is known for groups of rank at most two, yet not for general groups of rank three or higher. A first goal is to present the proofs for groups of rank at most two. There are two main parts. First, arguments based on counting constellations of zero-sum subsequences, which typically are obtained via group-ring or polynomial methods. These yield result for p-groups. Second, inductive arguments, which allow to obtain insights on zero-sum constants for a group G from knowledge of zero-sum constants for a subgroup H and the quotient group G/H. A second goal is to review further results to obtain a good overview of what is currently known on these constants.
Additional subjects to be discussed are the analogous problems for sets instead of sequences, the analogous problems for weighted sequences, and (time permitting) some applications.
Tentative schedule and selected references
1. Introduction and overview
2. Davenport constant for p-groups, related results (towards a proof of Kemnitz conjecture)
3. Proof of Kemnitz conjecture and inductive arguments
4. Some recent advances on zero-sum constants
5. Problems with weights
6. Set-based problems (Olson constant, Harborth constant)
7. Applications, summary, perspectives.
Selected references:
1. S. D. Adhikari, Plus-minus weighted zero-sum constans: a survey. In:Analytic Number Theory, Modular Forms and q-Hypergeometric Series (G. E. Andrews and F. Garvan, eds), Springer 2018.
2. W. D. Gao and A. Geroldinger, Zero-sum problems in finite abelian groups:a survey. Expo. Math. 24 (2006), 337–369.
3. A. Geroldinger, Additive group theory and non-unique factorizations. In:Combinatorial Number Theory and Additive Group Theory (A. Geroldinger and I. Z. Ruzsa, eds), Birkh ̈auser 2009.
Thursday, 27 February 2020
We present a classical theorem of Schur, also predating Ramsey Theory, regarding the existence of monochromatic solutions to the equation x+y-z=0. We then take up Rado’s theorem, a generalization to systems of Diophantine linear equations, and related questions.
In this lecture, we introduce the notion of additive dimension of a sumset $A+B$ and then focus on results specialized to two-dimensional sumsets, including compression techniques and discrete analogs of the Brunn-Minkowski Theorem.
We begin with examples of typical questions in sieve theory (such as twin primes and Goldbach problem). These will lead to a general formulation of the main tasks in sieve theory. Then we study the Selberg upper bound sieve, whose basic idea is amazingly simple (as compared to Brun's sieve, for example). We apply it to particular sieve problems (such as the ones mentioned above). At the end of this lecture series we describe how Selberg's method can be turned into a lower bound sieve.
Literature: "An Introduction to Sieve Methods and their Applications" by A.C. Cojucaru and R. Murty, "Opera de Cribro" by J. Friedlander and H. Iwaniec, "Sieve Methods" by H. Halberstam and H.E. Richert.
Sumfree sets in [1, N]: size of extremal sumfree sets, structure of large sumfree sets, number of sumfree sets.
Friday, 28 February 2020
We present some recent results concerning the degree of regularity of certain specific Diophantine equations of degree 1 and 2.
This lecture will be used to supplement the topics from the previous lectures as time permits.
We begin with examples of typical questions in sieve theory (such as twin primes and Goldbach problem). These will lead to a general formulation of the main tasks in sieve theory. Then we study the Selberg upper bound sieve, whose basic idea is amazingly simple (as compared to Brun's sieve, for example). We apply it to particular sieve problems (such as the ones mentioned above). At the end of this lecture series we describe how Selberg's method can be turned into a lower bound sieve.
Literature: "An Introduction to Sieve Methods and their Applications" by A.C. Cojucaru and R. Murty, "Opera de Cribro" by J. Friedlander and H. Iwaniec, "Sieve Methods" by H. Halberstam and H.E. Richert.
Sumfree sets in [1, N]: size of extremal sumfree sets, structure of large sumfree sets, number of sumfree sets
Saturday, 29 February 2020
We present some recent results concerning the degree of regularity of certain specific Diophantine equations of degree 1 and 2.
Addition in Z/pZ: Cauchy-Davenport, small doubling sets. Small doubling sets in Z/nZ.
Monday, 02 March 2020
The purpose of this series of lectures is to give an overview of zero-sum problems starting from classical results to questions of current research. A basic problem in zero-sum theory is the following problem: given a finite abelian group (G, +, 0) what is the smallest l such that each sequence of l elements over G has a non-empty subsequence whose terms sum to 0. This l is called the Davenport constant of G, denoted D(G). A classical variant of this problem is obtained by imposing a condition on the length of the subsequences:
given a finite abelian group (G, +, 0) what is the smallest l such that each sequence of l elements over G has a subsequence of length exp(G) whose terms sum to 0. This l is called the Erd ̋os–Ginzburg–Ziv constant of G, denoted s(G). The value of both constants is known for groups of rank at most two, yet not for general groups of rank three or higher. A first goal is to present the proofs for groups of rank at most two. There are two main parts. First, arguments based on counting constellations of zero-sum subsequences, which typically are obtained via group-ring or polynomial methods. These yield result for p-groups. Second, inductive arguments, which allow to obtain insights on zero-sum constants for a group G from knowledge of zero-sum constants for a subgroup H and the quotient group G/H. A second goal is to review further results to obtain a good overview of what is currently known on these constants.
Additional subjects to be discussed are the analogous problems for sets instead of sequences, the analogous problems for weighted sequences, and (time permitting) some applications.
Tentative schedule and selected references
1. Introduction and overview
2. Davenport constant for p-groups, related results (towards a proof of Kemnitz conjecture)
3. Proof of Kemnitz conjecture and inductive arguments
4. Some recent advances on zero-sum constants
5. Problems with weights
6. Set-based problems (Olson constant, Harborth constant)
7. Applications, summary, perspectives.
Selected references:
1. S. D. Adhikari, Plus-minus weighted zero-sum constans: a survey. In:Analytic Number Theory, Modular Forms and q-Hypergeometric Series(G. E. Andrews and F. Garvan, eds), Springer 2018.
2. W. D. Gao and A. Geroldinger, Zero-sum problems in finite abelian groups:a survey. Expo. Math. 24 (2006), 337–369.
3. A. Geroldinger, Additive group theory and non-unique factorizations. In:Combinatorial Number Theory and Additive Group Theory (A. Geroldinger and I. Z. Ruzsa, eds), Birkh ̈auser 2009.
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We begin with examples of typical questions in sieve theory (such as twin primes and Goldbach problem). These will lead to a general formulation of the main tasks in sieve theory. Then we study the Selberg upper bound sieve, whose basic idea is amazingly simple (as compared to Brun's sieve, for example). We apply it to particular sieve problems (such as the ones mentioned above). At the end of this lecture series we describe how Selberg's method can be turned into a lower bound sieve.
Literature: "An Introduction to Sieve Methods and their Applications" by A.C. Cojucaru and R. Murty, "Opera de Cribro" by J. Friedlander and H. Iwaniec, "Sieve Methods" by H. Halberstam and H.E. Richert.
Sumfree subsets of abelian groups.
Tuesday, 03 March 2020
The purpose of this series of lectures is to give an overview of zero-sum problems starting from classical results to questions of current research. A basic problem in zero-sum theory is the following problem: given a finite abelian group (G, +, 0) what is the smallest l such that each sequence of l elements over G has a non-empty subsequence whose terms sum to 0. This l is called the Davenport constant of G, denoted D(G). A classical variant of this problem is obtained by imposing a condition on the length of the subsequences:
given a finite abelian group (G, +, 0) what is the smallest l such that each sequence of l elements over G has a subsequence of length exp(G) whose terms sum to 0. This l is called the Erd ̋os–Ginzburg–Ziv constant of G, denoted s(G). The value of both constants is known for groups of rank at most two, yet not for general groups of rank three or higher. A first goal is to present the proofs for groups of rank at most two. There are two main parts. First, arguments based on counting constellations of zero sum subsequences, which typically are obtained via group-ring or polynomial methods. These yield result for p-groups. Second, inductive arguments, which allow to obtain insights on zero-sum constants for a group G from knowledge of zero-sum constants for a subgroup H and the quotient group G/H. A second goal is to review further results to obtain a good overview of what is currently known on these constants.
Additional subjects to be discussed are the analogous problems for sets instead of sequences, the analogous problems for weighted sequences, and (time permitting) some applications.
Tentative schedule and selected references
1. Introduction and overview
2. Davenport constant for p-groups, related results (towards a proof of Kemnitz conjecture)
3. Proof of Kemnitz conjecture and inductive arguments
4. Some recent advances on zero-sum constants
5. Problems with weights
6. Set-based problems (Olson constant, Harborth constant)
7. Applications, summary, perspectives.
Selected references:
1. S. D. Adhikari, Plus-minus weighted zero-sum constans: a survey. In:Analytic Number Theory, Modular Forms and q-Hypergeometric Series(G. E. Andrews and F. Garvan, eds), Springer 2018.
2. W. D. Gao and A. Geroldinger, Zero-sum problems in finite abelian groups:a survey. Expo. Math. 24 (2006), 337–369.
3. A. Geroldinger, Additive group theory and non-unique factorizations. In:Combinatorial Number Theory and Additive Group Theory (A. Geroldinger and I. Z. Ruzsa, eds), Birkh ̈auser 2009.
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Structure of large sumfree sets in Z/pZ: from the first approach of V. Lev to the most recent results.
Wednesday, 04 March 2020
The purpose of this series of lectures is to give an overview of zero-sum problems starting from classical results to questions of current research. A basic problem in zero-sum theory is the following problem: given a finite abelian group (G, +, 0) what is the smallest l such that each sequence of l elements over G has a non-empty subsequence whose terms sum to 0. This l is called the Davenport constant of G, denoted D(G). A classical variant of this problem is obtained by imposing a condition on the length of the subsequences:
given a finite abelian group (G, +, 0) what is the smallest l such that each sequence of l elements over G has a subsequence of length exp(G) whose terms sum to 0. This l is called the Erd ̋os–Ginzburg–Ziv constant of G, denoted s(G). The value of both constants is known for groups of rank at most two, yet not for general groups of rank three or higher. A first goal is to present the proofs for groups of rank at most two. There are two main parts. First, arguments based on counting constellations of zero-sum subsequences, which typically are obtained via group-ring or polynomial methods. These yield result for p-groups. Second, inductive arguments, which allow to obtain insights on zero-sum constants for a group G from knowledge of zero-sum constants for a subgroup H and the quotient group G/H. A second goal is to review further results to obtain a good overview of what is currently known on these constants.
Additional subjects to be discussed are the analogous problems for sets instead of sequences, the analogous problems for weighted sequences, and (time permitting) some applications.
Tentative schedule and selected references
1. Introduction and overview
2. Davenport constant for p-groups, related results (towards a proof of Kemnitz conjecture)
3. Proof of Kemnitz conjecture and inductive arguments
4. Some recent advances on zero-sum constants
5. Problems with weights
6. Set-based problems (Olson constant, Harborth constant)
7. Applications, summary, perspectives.
Selected references:
1. S. D. Adhikari, Plus-minus weighted zero-sum constans: a survey. In: Analytic Number Theory, Modular Forms and q-Hypergeometric Series. (G. E. Andrews and F. Garvan, eds), Springer 2018.
2. W. D. Gao and A. Geroldinger, Zero-sum problems in finite abelian groups: a survey. Expo. Math. 24 (2006), 337–369.
3. A. Geroldinger, Additive group theory and non-unique factorizations. In: Combinatorial Number Theory and Additive Group Theory (A. Geroldinger and I. Z. Ruzsa, eds), Birkh ̈auser 2009.
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Structure of large sumfree sets in Z/pZ: from the first approach of V. Lev to the most recent results
Thursday, 05 March 2020
The purpose of this series of lectures is to give an overview of zero-sum problems starting from classical results to questions of current research. A basic problem in zero-sum theory is the following problem: given a finite abelian group (G, +, 0) what is the smallest l such that each sequence of l elements over G has a non-empty subsequence whose terms sum to 0. This l is called the Davenport constant of G, denoted D(G). A classical variant of this problem is obtained by imposing a condition on the length of the subsequences:
given a finite abelian group (G, +, 0) what is the smallest l such that each sequence of l elements over G has a subsequence of length exp(G) whose terms sum to 0. This l is called the Erd ̋os–Ginzburg–Ziv constant of G, denoted s(G). The value of both constants is known for groups of rank at most two, yet not for general groups of rank three or higher. A first goal is to present the proofs for groups of rank at most two. There are two main parts. First, arguments based on counting constellations of zero-sum subsequences, which typically are obtained via group-ring or polynomial methods. These yield result for p-groups. Second, inductive arguments, which allow to obtain insights on zero-sum constants for a group G from knowledge of zero-sum constants for a subgroup H and the quotient group G/H. A second goal is to review further results to obtain a good overview of what is currently known on these constants.
Additional subjects to be discussed are the analogous problems for sets instead of sequences, the analogous problems for weighted sequences, and (time permitting) some applications.
Tentative schedule and selected references
1. Introduction and overview
2. Davenport constant for p-groups, related results (towards a proof of Kemnitz conjecture)
3. Proof of Kemnitz conjecture and inductive arguments
4. Some recent advances on zero-sum constants
5. Problems with weights
6. Set-based problems (Olson constant, Harborth constant)
7. Applications, summary, perspectives.
Selected references:
1. S. D. Adhikari, Plus-minus weighted zero-sum constans: a survey. In: Analytic Number Theory, Modular Forms and q-Hypergeometric Series (G. E. Andrews and F. Garvan, eds), Springer 2018.
2. W. D. Gao and A. Geroldinger, Zero-sum problems in finite abelian groups: a survey. Expo. Math. 24 (2006), 337–369.
3. A. Geroldinger, Additive group theory and non-unique factorizations. In: Combinatorial Number Theory and Additive Group Theory (A. Geroldinger and I. Z. Ruzsa, eds), Birkh ̈auser 2009.
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Friday, 06 March 2020
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We shall discuss two-term supercongruences for Apéry-like numbers.
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