In this paper, we extend the definition of the Nathanson height from points in projective spaces over F p to points in projective spaces over arbitrary finite fields. If OEa 0 W : : : W a n 2 P d .F p /, then the Nathanson height iswhere H.a i / D jN.a i /j C p.deg.a i / 1/ with N the field norm and jN.a i /j the element of ¹0; 1; : : : ; p 1º congruent to N.a i / modulo p.We investigate the basic properties of this extended height, provide some bounds, study its image on the projective line h p .P 1 .F p // and propose some questions for further research.

a b s t r a c tIn this paper, we develop a Helly-Gallai type theorem for piercing number on finite families of closed intervals in R d , as well as some bounds for the piercing number of families of lines and intervals satisfying the (p, 3)-property.

No abstract

We say that a set of the form [Formula: see text] for some [Formula: see text] is an interval. For a nonempty finite subset [Formula: see text] of [Formula: see text] and [Formula: see text], Vsevolod Lev proved in [Optimal representations by sumsets and subset sums, J. Number Theory 62(1) (1997) 127–143] some results about the existence of long intervals contained in the [Formula: see text]-fold iterated sumset of [Formula: see text]. Furthermore, in the same paper, he proposed a conjecture [Optimal representations by sumsets and subset sums, J. Number Theory 62(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.

Let X and Y be nonempty finite subsets of 𝕑 and X +Y its sumset. The structures of X and Y when r(X, Y ):= |X +Y |−|X|−|Y | is small have been widely studied; in particular the Generalized Freiman’s 3k − 4 Theorem describes X and Y when r(X, Y ) ≤ min{|X|, |Y |} − 4. However, not too much is known about X and Y when r(X, Y ) > min{|X|, |Y |} − 4. In this paper we study the structure of X and Y for arbitrary r(X, Y ).