Restricted sumset
In additive number theory and combinatorics, a restricted sumset has the form S = { a 1 + ⋯ + a n : a 1 ∈ A 1 , … , a n ∈ A n a n d P ( a 1 , … , a n ) ≠ 0 } , {\displaystyle S=\{a_{1}+\cdots +a_{n}:\ a_{1}\in A_{1},\ldots ,a_{n}\in A_{n}\ \mathrm {and} \ P(a_{1},\ldots ,a_{n})\not =0\},} where A 1 , … , A n {\displaystyle A_{1},\ldots ,A_{n}} are finite nonempty subsets of a field F and P ( x 1 , … , x n ) {\displaystyle P(x_{1},\ldots ,x_{n})} is a polynomial over F. If P {\displaystyle P} is a constant non-zero function, for example P ( x 1 , … , x n ) = 1 {\displaystyle P(x_{1},\ldots ,x_{n})=1} for any x 1 , … , x n {\displaystyle x_{1},\ldots ,x_{n}} , then S {\displaystyle S} is the usual sumset A 1 + ⋯ + A n {\displaystyle A_{1}+\cdots +A_{n}} which is denoted by n A {\displaystyle nA} if A 1 = ⋯ = A n = A . {\displaystyle A_{1}=\cdots =A_{n}=A.} When P ( x 1 , … , x n ) = ∏ 1 ≤ i < j ≤ n ( x j − x i ) , {\displaystyle P(x_{1},\ldots ,x_{n})=\prod _{1\leq i<j\leq n}(x_{j}-x_{i}),} S is written as A 1 ∔ ⋯ ∔ A n {\displaystyle A_{1}\dotplus \cdots \dotplus A_{n}} which is denoted by n ∧ A {\displaystyle n^{\wedge }A} if A 1 = ⋯ = A n = A .