Narrow Results

A finitely generated ℤ[t, t^-1]-module without ℤ-torsion and having nonzero order Δ(M) of degree d is determined by a pair of sub-lattices of ℤ^d. Their indices are the absolute values of the leading and trailing coefficients of Δ(M). This description has applications in knot theory.

Starting from a result of Stewart, Tijdeman and Ruzsa on iterated difference sequences, we introduce the notion of iterated compositions of linear operations. We prove a general result on the stability of such compositions (with bounded coefficients) on sets of integers having a positive upper density.

We say that a set of the form $[a,b]:=\{c\in \mathbb{Z}:a\le c\le b\}$ for some $a,b\in \mathbb{Z}$ is an interval. For a nonempty finite subset $A$ of $\mathbb{Z}$ and $n\in \mathbb{N}$, Vsevolod Lev proved in [Optimal representations by sumsets and subset sums, J. Number Theory62(1) (1997) 127–143] some results about the existence of long intervals contained in the $n$-fold iterated sumset of $A$. Furthermore, in the same paper, he proposed a conjecture [Optimal representations by sumsets and subset sums, J. Number Theory62(1) (1997) 127–143, Conjecture 1], see also [V. Lev, Consecutive integers in high-multiplicity sumsets, Acta Math. Hungar.129(3) (2010) 245–253, Conjecture 1]. Lev proved some particular cases of his conjecture, and he showed that these few cases have important applications. In this paper, we prove his conjecture.