Cyclic Spaces for Grassmann Derivatives and Additive Theory

Abstract

Let A be a finite subset of Zp (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∈Zp 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 operatorfon V, restricted to the mth Grassmann space ∧mV. 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.

Combinatorial Nullstellensatz

“…The following result of Dias da Silva and Hamidoune [21] is a simple consequence of (a special case of) the above theorem. Proof.…”

“…This easily implies the following theorem, conjectured by Erdős and Heilbronn in 1964 (cf., e.g., [25]). Special cases of this conjecture have been proved by various researchers ( [49], [43], [50], [29]) and the full conjecture has recently been proved by Dias Da Silva and Hamidoune [21], using some tools from linear algebra and the representation theory of the symmetric group.…”

Long arithmetic progressions in sumsets: Thresholds and bounds

“…This is what Erdös and Heilbronn conjectured. While CauchyDavenport's theorem is quite easy to prove, Erdös-Heilbronn's conjecture had been open for about thirty years until it was solved by da Silva and Hamidoune in 1994 [7]. It is now not so big a surprise that Theorem 7.1 is harder and deeper than both Theorem 3.12 and Theorem 5.1.…”

Secure Communication of Local States in Interpreted Systems

“…An announcement is (card) safe if (1) and (2) hold. To find out when this is the case, we proceed combinatorially, building on results of [3] and [9]. This gives us the following Corollary 3.…”