Small points on subvarieties of a torus

Abstract

Abstract. Let V be a subvariety of a torus defined over the algebraic numbers. We give a qualitative and quantitative description of the set of points of V of height bounded by invariants associated to any variety containing V . Especially, we determine whether such a set is or not dense in V . We then prove that these sets can always be written as the intersection of V with a finite union of translates of tori of which we control the sum of the degrees.As a consequence, we prove a conjecture by the first author and David up to a logarithmic factor.

On the length of arithmetic progressions in linear combinations of S-units

“…The first one is due to Amoroso and Viada [1] and concerns the number of nondegenerate solutions to linear equations with variables from Γ. This result is a recent improvement of a result from [6].…”

On permutations of Hardy-Littlewood-Pólya sequences

“…Specifically, we will establish an "interlaced" mixing property for normed sums of {n k x}, expressed by Lemmas 4 and 6, implying that the sequence {n k x} has mixing properties after any permutation of its terms. This property is considerably stronger than standard weak dependence properties of lacunary series, which are typically valid only in the natural order of elements and will be deduced from profound Diophantine properties of (n k ) proved recently by Amoroso and Viada [1]. As a consequence, we obtain that for any permutation σ : N → N of the positive integers k≤N f (n σ(k) x) obeys, under mild conditions on the periodic function f , the CLT and LIL, except that the norming sequence depends strongly on the permutation σ. Computing the norming factors in the permuted CLT and LIL is generally a difficult number theoretical problem; examples will be given below and in Section 3.…”

“…. , q r be a fixed set of coprime integers and let (n k ) be the set of numbers q α 1 1 · · · q α r r , (α i ≥ 0 integers) arranged in increasing order. Such sequences are (sometimes) called Hardy-Littlewood-Pólya sequences, and their distribution has been investigated extensively in number theory.…”

Successive minima of toric height functions

“…We consider its universal line bundle O P(E) (1), that is the line bundle corresponding to the Cartier divisor D := a 0 D 0 + D 1 , where D 0 denotes the inverse image in P(E) of the hyperplane at infinity of P n and…”

“…The effective version of the generalized Bogomolov conjecture asks for an explicit lower bound for the essential minimum of certain varieties in terms of geometric and arithmetic data. Such lower bounds have been extensively studied and have several applications in Diophantine geometry and computer algebra, see for instance [2,1].…”