Cyclic Spaces for Grassmann Derivatives and Additive Theory |
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Authors: | Da Silva J A Dias; Hamidoune Y O |
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Institution: | Universidade de Lisboa, Rua Ernesto de Vasconcelos Bloco Cl, 3° Piso, 1700 Lisboa, Portugal
Université Pierre et Marie Curie E. Combinatoire, UFR 921, 4 Place Jussieu, 75005 Paris, France |
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Abstract: | Let A be a finite subset of Zp (where p is a prime). Erdösand Heilbronn conjectured (1964) that the set of sums of the2-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 Ahas cardinality at least min {p,m(|A| m)+ 1}. In particular,we answer affirmatively the above conjecture. We apply thisresult to the problem of finding the smallest n such that forevery subset 5 of cardinality n and every xZp there is a subsetof S with sum equal to x. On this last problem we improve theknown results due to Erdös and Heilbronn and to Olson. The above result will be derived from the following generalproblem on Grassmann spaces. Let F be a field and let V be afinite dimensional vector space of dimension d over F. Let pbe the characteristic of F in nonzero characteristic and otherwise. Let Df be the derivative of a linear operatorfon V, restrictedto the mth Grassmann space mV. We show that there is a cyclicsubspace for the derivative with dimension at least min {p,m(nm)+ 1}, where n is the maximum dimension of the cyclic subspacesof f. This bound is sharp and is reached when f has d distincteigenvalues forming an arithmetic progression. |
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