Glossary

1.2 Basic definitions and notation

4 min read•july 30, 2024

Additive combinatorics blends number theory and combinatorics, focusing on the structure of sets under addition. This field explores how sets grow when combined and studies patterns in sumsets, laying the groundwork for understanding additive properties of integers and finite fields.

Basic definitions and notation are crucial in additive combinatorics. We'll cover essential concepts like sets, subsets, and their properties, as well as key operations like sumsets and difference sets. These fundamentals will help us tackle more complex problems in the field.

Essential terms in additive combinatorics

Sets, subsets, and their properties

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  • A set is a well-defined collection of distinct objects
  • A subset is a set that is contained within another set
  • The cardinality of a set refers to the number of elements in the set, denoted by [∣A∣](https://www.fiveableKeyTerm:∣a∣)[|A|](https://www.fiveableKeyTerm:|a|) for a set AA
  • The complement of a set AA, denoted AcA^c, is the set of all elements in the universal set that are not in AA
  • The symmetric difference of two sets AA and BB, denoted A△BA \triangle B, is the set of elements that are in either AA or BB, but not in both

Sums and differences of sets

  • The sum of two sets AA and BB, denoted [A + B](https://www.fiveableKeyTerm:a_+_b), is the set of all elements that can be expressed as the sum of an element from AA and an element from BB
  • The h-fold sum of a set AA, denoted hAhA, is the set of all elements that can be expressed as the sum of hh elements from AA
  • The difference set of a set AA, denoted [A - A](https://www.fiveableKeyTerm:a_-_a), is the set of all differences between pairs of elements in AA
    • For example, if A={1,2,3}A = \{1, 2, 3\}, then A−A={−2,−1,0,1,2}A - A = \{-2, -1, 0, 1, 2\}

Mathematical notation in additive combinatorics

Set builder and interval notation

  • Set builder notation, such as {x∈A:P(x)}\{x \in A : P(x)\}, represents the set of all elements xx in AA that satisfy the condition P(x)P(x)
    • For example, {x∈N:x is even}\{x \in \mathbb{N} : x \text{ is even}\} represents the set of all even natural numbers
  • Interval notation, such as [a,b][a, b] or (a,b)(a, b), represents the set of all real numbers between aa and bb, inclusive or exclusive of the endpoints, respectively
    • For example, [0,1][0, 1] represents the set of all real numbers between 0 and 1, including 0 and 1

Set operations and products

  • The union of two sets AA and BB, denoted A∪BA \cup B, is the set of all elements that are in either AA or BB, or both
  • The intersection of two sets AA and BB, denoted A∩BA \cap B, is the set of all elements that are in both AA and BB
  • The Cartesian product of two sets AA and BB, denoted A×BA \times B, is the set of all ordered pairs (a,b)(a, b) where a∈Aa \in A and b∈Bb \in B
  • The power set of a set AA, denoted P(A)\mathcal{P}(A), is the set of all subsets of AA, including the empty set and AA itself

Types of sets and their properties

Finite and infinite sets

  • A is a set with a finite number of elements, while an has an infinite number of elements
    • For example, the set of natural numbers N\mathbb{N} is an infinite set, while the set {1,2,3}\{1, 2, 3\} is a finite set
  • Countable sets are infinite sets that can be put into a one-to-one correspondence with the natural numbers, while uncountable sets cannot
    • The set of rational numbers Q\mathbb{Q} is countable, while the set of real numbers R\mathbb{R} is uncountable

Structured sets and their axioms

  • Structured sets, such as groups, rings, and fields, have additional properties and operations defined on them that satisfy certain axioms
  • A group is a set with a binary operation that satisfies the group axioms: closure, associativity, identity, and inverses
    • For example, the set of integers Z\mathbb{Z} under addition forms a group
  • A ring is a set with two binary operations (addition and multiplication) that satisfy the ring axioms, which include the group axioms for addition and distributivity of multiplication over addition
    • For example, the set of integers Z\mathbb{Z} under addition and multiplication forms a ring
  • A field is a ring in which every non-zero element has a multiplicative inverse
    • For example, the set of real numbers R\mathbb{R} under addition and multiplication forms a field

Set addition in additive combinatorics

Definition and properties of set addition

  • Set addition is a fundamental operation in additive combinatorics that combines elements from two or more sets to create a new set
  • The sum set A+BA + B contains all possible sums of elements from AA and BB, i.e., {a+b:a∈A,b∈B}\{a + b : a \in A, b \in B\}
  • Set addition is commutative (A+B=B+AA + B = B + A) and associative ((A+B)+C=A+(B+C)(A + B) + C = A + (B + C))

h-fold sums and repeated sums

  • The h-fold sum of a set AA, denoted hAhA, is the set of all possible sums of hh elements from AA, i.e., {a1+a2+...+ah:ai∈A for all i}\{a_1 + a_2 + ... + a_h : a_i \in A \text{ for all } i\}
    • For example, if A={1,2}A = \{1, 2\}, then 2A={2,3,4}2A = \{2, 3, 4\}
  • The repeated sum of a set AA, denoted A+A+...+AA + A + ... + A (hh times), is equal to the h-fold sum hAhA
  • Set addition is used to study the additive structure of sets and to prove important results in additive combinatorics, such as the and Freiman's theorem

Key Terms to Review (18)

|a|: |a| represents the absolute value of a number 'a', indicating its distance from zero on the number line without considering its sign. This concept is crucial as it helps in analyzing mathematical relationships and inequalities, providing a measure of magnitude that is essential in various mathematical contexts, including equations and inequalities.
A - a: The notation 'a - a' represents the concept of the difference between an element and itself, which is always equal to zero. This simple yet fundamental expression is often encountered in various mathematical contexts, including algebra and combinatorics, where understanding the significance of zero and its properties is crucial for deeper explorations of equations and set operations.
A + b: In the context of additive combinatorics, 'a + b' represents the addition operation between two elements, typically integers or real numbers. This operation is fundamental as it forms the basis for exploring various properties of sums and combinations, allowing for a deeper understanding of numerical relationships and structures.
A ⊆ b: The notation 'a ⊆ b' signifies that set 'a' is a subset of set 'b', meaning every element of 'a' is also an element of 'b'. This relationship is fundamental in set theory, allowing for comparisons and operations involving different sets, while establishing the idea of inclusion and hierarchy among sets.
Additive Group: An additive group is a mathematical structure consisting of a set equipped with an operation of addition that satisfies certain properties, namely closure, associativity, identity, and invertibility. This concept is essential in understanding the foundations of additive combinatorics, as it provides a framework for analyzing how elements combine within specific sets and underpins many theorems and results in the field.
Catalan Numbers: Catalan numbers are a sequence of natural numbers that have significant applications in combinatorial mathematics, defined by the formula $$C_n = \frac{1}{n+1} \binom{2n}{n}$$. They count various combinatorial structures, such as the number of valid parentheses combinations, the number of ways to connect points in a plane without intersecting lines, and the number of rooted binary trees with a given number of leaves.
Cauchy-Davenport Theorem: The Cauchy-Davenport Theorem states that if $A$ and $B$ are finite subsets of the cyclic group $\mathbb{Z}/p\mathbb{Z}$, where $p$ is a prime number, then the size of the sumset $A + B$ is at least $\min(p, |A| + |B| - 1)$. This theorem is a fundamental result in additive combinatorics that links the sizes of sets and their sums in modular arithmetic.
Characteristic Function: The characteristic function is a mathematical tool that encapsulates the distribution of a random variable by representing it as a Fourier transform. It connects probability distributions with functional analysis, allowing us to analyze properties like moments and convergence of random variables. This function is crucial in probability theory and statistics as it helps in identifying distributions and proving limit theorems.
D-dimensional progression: A d-dimensional progression is a generalization of arithmetic progressions in multiple dimensions, defined as a set of points that can be represented as $$\{(a_1 + kd_1, a_2 + kd_2, \ldots, a_d + kd_d) : k \in \mathbb{Z}\}$$ for some integers $$a_i$$ and non-zero integers $$d_i$$. This concept extends the idea of linear sequences to higher dimensions, allowing for the exploration of patterns and relationships among numbers in a multi-dimensional space.
Density: In additive combinatorics, density refers to the proportion of integers in a given set relative to the whole set of natural numbers. It serves as a critical measure of how 'thick' or 'thin' a subset of numbers is within the larger context, influencing various results and theorems in the field.
Erdős–Ginzburg–Ziv Theorem: The Erdős–Ginzburg–Ziv theorem states that for any set of $2n-1$ integers, there exists a subset of $n$ integers whose sum is divisible by $n$. This theorem highlights the fundamental interplay between combinatorics and number theory, and it serves as a key result in additive combinatorics.
Finite set: A finite set is a collection of distinct elements that has a limited number of members, meaning that you can count the elements in it and reach a specific total. The concept of finite sets is fundamental in mathematics, particularly in combinatorics and set theory, where understanding the size and properties of these sets plays a crucial role in various calculations and proofs.
Indicator Function: An indicator function is a mathematical function that assigns a value of 1 to elements of a specific subset and 0 to all other elements in the universal set. This function is useful in various areas, particularly in combinatorics and analysis, as it helps simplify expressions by clearly defining subsets and their properties. In relation to additive combinatorics, indicator functions play a crucial role in analyzing sets and their interactions, particularly in proofs and theorems.
Infinite set: An infinite set is a collection of distinct elements that has no end or limit, meaning it can be put into a one-to-one correspondence with a proper subset of itself. This concept is crucial for understanding various types of infinity, as infinite sets can be countably or uncountably infinite, which leads to significant implications in mathematics. The ability to categorize infinite sets forms the foundation for exploring cardinality and the nature of mathematical structures.
K-term progression: A k-term progression is a sequence of numbers where each term can be expressed as a linear combination of the previous terms, with constant differences between them. This concept is pivotal in additive combinatorics, as it helps analyze patterns within sets of numbers and their arithmetic structures. Understanding k-term progressions provides insight into more complex mathematical phenomena, such as the relationships between subsets of integers and their behaviors under various operations.
Sidon Set: A Sidon set is a subset of integers such that all pairwise sums of its elements are distinct. In other words, for a set \( S \), if you take any two elements \( a \) and \( b \) from \( S \), the sum \( a + b \) must be different from the sum of any other pair of elements in the set. This unique property connects the concept to additive combinatorics and has implications for understanding the structure of number sets.
Sum-free set: A sum-free set is a subset of integers such that no two elements in the set can be added together to form another element in the same set. This concept connects to various areas in combinatorics, particularly in understanding the structure and properties of sets of integers. Sum-free sets have significant implications in additive number theory and can be related to the study of higher-order additive properties, such as those explored in Gowers norms and more complex combinatorial configurations.
Sumset: A sumset is the set formed by adding every element of one set to every element of another set. This concept is foundational in additive combinatorics, as it helps explore relationships between sets and their additive properties. Understanding sumsets lays the groundwork for more advanced theories, including the behavior of finite groups and the implications of various combinatorial results.
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