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Cyclic spaces for Grassmann derivatives and additive theory. (English) Zbl 0819.11007

Summary: Let \(A\) be a finite subset of \(\mathbb{Z}_ p\) (where \(p\) is a prime). P. Erdős and H. Heilbronn [Acta Arith. 9, 149-159 (1964; Zbl 0156.04801)] conjectured 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 \(S\) of cardinality \(n\) and every \(x\in Z_ p\) there is a subset of \(S\) with sum equal to \(x\). On this last problem we improve the known results due to Erdős 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 \(\infty\) otherwise. Let \(Df\) be the derivative of a linear operator \(f\) on \(V\), restricted to the \(m\)-th Grassmann space \(\wedge^ m V\). 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.

MSC:

11B75 Other combinatorial number theory
15A69 Multilinear algebra, tensor calculus
20D60 Arithmetic and combinatorial problems involving abstract finite groups

Citations:

Zbl 0156.04801
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