Cauchy–Davenport theorem

For nonempty subsets A and B of the integers modulo a prime p, their sumset has at least min(p, |A| + |B| − 1) elements.

Sumset: The set of all sums formed by adding one element from each of two sets. The theorem bounds the size of this operation in the prime cyclic group.
Augustin-Louis Cauchy: A French mathematician whose work ranged across analysis, algebra, and number theory. His early form of the sumset bound is one source of the theorem's name.
Cauchy–Davenport theorem for multiple summands: The iterated bound stating that a sum of k nonempty subsets modulo a prime has size at least min(p, sum of their sizes minus k plus 1). Repeated application turns the two-set inequality into a bound for many summands.
Kneser’s theorem: A theorem describing lower bounds for sumsets in abelian groups through their periods. For composite moduli, nontrivial periods can invalidate the Cauchy–Davenport bound.
Combinatorial Nullstellensatz: A theorem connecting nonzero polynomial coefficients with the existence of points in finite Cartesian products. Its coefficient method yields a short proof of Cauchy–Davenport.
Harold Davenport: A British number theorist known for work in analytic number theory and additive problems. His independent proof established the result in the form commonly associated with the theorem.
Restricted sumset: A sumset in which selected combinations, often repeated summands, are excluded. The theorem's growth principle motivates sharper bounds when summands must differ.
Cauchy–Davenport theorem for composite moduli: Sumset bounds for residues modulo a composite integer, where subgroup structure can limit growth. The prime-modulus conclusion does not hold unchanged when the modulus is composite.
Polynomial method: A proof strategy that encodes a discrete problem in a polynomial and exploits its algebraic properties. Cauchy–Davenport is a classic application of this strategy.
Additive number theory: The study of arithmetic questions about sums of integers and other algebraic objects. Cauchy–Davenport became a foundational result in this area.

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Additive combinatoricsRelated: It is a foundational example of a sharp lower bound on sumset growth.