. Let p be a prime and let A = ( a 1 ,...,a ℓ ) be a sequence of nonzero elements in F p . In this paper, we study the set of all 0-1 solutions to the equation a 1 x 1 + ··· + a ℓ x ℓ = 0 . We prove that whenever ℓ > p , this set actually characterizes A up to a nonzero multiplicative constant, which is no longer true for ℓ < p . The critical case ℓ = p is of particular interest. In this context, we prove that whenever ℓ = p and A is nonconstant, the above equation has at least p − 1 minimal 0… Expand

For a sequence $S$ of terms from an abelian group $G$ of length $|S|$, let $\Sigma_n(S)$ denote the set of all elements that can be represented as the sum of terms in some $n$-term subsequence of… Expand

THEOREM. Each set of 2n-1 integers contains some subset of n elements the sum of which is a multiple of n. PROOF. Assume first n = p (p prime). Our theorem is trivial for p = 2, thus henceforth p >… Expand

Using the polynomial method in additive number theory, this article establishes a new addition theorem for the set of subsums of a set satisfying $A\cap(-A)=\emptyset$ in $\mathbb{Z}/p\mathbb{Z}$:… Expand

The following theorem is proved. Ifa1,a2, ...an are nonzero elements inZn, and are not all equal, then ε1a1+ε2a2+...+εnan=0 has at leastn solutions with εi=0 or 1.

CONCEPTS IN FACTORIZATION THEORY AND EXAMPLES Atoms and Primes Free Monoids, Factorial Monoids and Factorizations BF-Monoids Systems of Sets of Lengths FF-Monoids The Catenary Degree and the Tame… Expand