Sumset

A sumset, denoted as A + B, is a set of all possible sums of pairs of elements taken from two sets, A and B. More formally, if A and B are sets, then their sumset is defined as:

A + B = {a + b : a ∈ A, b ∈ B}

The elements in A and B can belong to any algebraic structure where addition is defined, such as integers, real numbers, vectors, or elements of a group.

Properties and Concepts

• Cardinality: The cardinality (number of elements) of the sumset A + B is at most |A| * |B|, where |A| and |B| denote the cardinalities of sets A and B, respectively. However, it can be smaller if there are duplicate sums. Determining the size of the sumset is a central problem in additive combinatorics.
• Difference Set: Related to the sumset is the difference set, denoted A - B, which is defined as {a - b : a ∈ A, b ∈ B}.
• Additive Combinatorics: Sumsets are a fundamental concept in additive combinatorics, a field of mathematics that studies the combinatorial properties of sets of numbers under addition and related operations. Problems in this area often involve finding lower or upper bounds on the size of sumsets, or characterizing sets with specific properties regarding their sumsets.
• Multiple Sumsets: We can also consider multiple sumsets such as kA = A + A + ... + A (k times), where each element of A is added to the other elements of A (including itself), repeated k times.
• Applications: Sumsets have applications in various fields, including number theory, harmonic analysis, and theoretical computer science. They are used, for example, in the study of arithmetic progressions and in the construction of error-correcting codes.
• Sidon Sets: A set A is called a Sidon set if all the sums a + b, where a, b ∈ A and a ≤ b, are distinct. Sidon sets are sets with small sumsets, in the sense that |A + A| = |A|(|A|+1)/2.