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Cyclic Spaces for Grassmann Derivatives and Additive Theory
Author(s) -
Da Silva J. A. Dias,
Hamidoune Y. O.
Publication year - 1994
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/26.2.140
Subject(s) - mathematics , cardinality (data modeling) , linear subspace , combinatorics , dimension (graph theory) , conjecture , vector space , space (punctuation) , finite field , eigenvalues and eigenvectors , subspace topology , prime (order theory) , discrete mathematics , pure mathematics , mathematical analysis , linguistics , philosophy , physics , quantum mechanics , computer science , data mining
Let A be a finite subset of Z p (where p is a prime). Erdös and Heilbronn conjectured (1964) that the set of sums of the 2‐subsets of A has cardinality at least min( p , 2|A| — 3). We show here that the set of sums of all m ‐subsets of A has cardinality at least min { p , m (|A| — m )+ 1}. In particular, we answer affirmatively the above conjecture. We apply this result to the problem of finding the smallest n such that for every subset 5 of cardinality n and every x ∈ Z p there is a subset of S with sum equal to x . On this last problem we improve the known results due to Erdös and Heilbronn and to Olson. The above result will be derived from the following general problem on Grassmann spaces. Let F be a field and let V be a finite dimensional vector space of dimension d over F . Let p be the characteristic of F in nonzero characteristic and ∞ otherwise. Let Df be the derivative of a linear operator f on V , restricted to the m th Grassmann space ∧ m V . We show that there is a cyclic subspace for the derivative with dimension at least min { p , m ( n − m ) + 1}, where n is the maximum dimension of the cyclic subspaces of f . This bound is sharp and is reached when f has d distinct eigenvalues forming an arithmetic progression.