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Consider the "points" to be in $\mathbb{G}_m(\mathbb{C}) = \mathbb{C}^{\times}$. By a sequence of "arithmetically small points" is meant a sequence of pairwise different algebraic points $\beta$ that, …

Consider, on the one hand, algebraic integers $\alpha$ and their rational approximants to within a varying exponent $\kappa > 2$; and on the other hand, smooth projective geometrically irreducible cur …

Previously I have mentioned the following problem in an addition to the list of Contest problems with connections to deeper mathematics. Is there an infinite bounded sequence $(P_n) \subset \mathbb{ …

Faltings, in the same paper (Diophantine approximation on abelian varieties, Ann. Math. 1991) in which he proved his famous ``big theorem,'' proved also that at any place $v$ of a number field $K$ and …

With the constant $1$, this is Minkowski's higher dimensional extension of Dirichlet's approximation theorem: If $\alpha_1, \ldots,\alpha_n$ are real numbers, then there are rationals $p_i/q$ with $| …

I believe it is the general opinion, at least among those working in diophantine approximations, that the extreme case of Salem numbers (the question of the title) would be just as difficult as the fu …

Let $f : \mathbb{CP}^1 \to \mathbb{CP}^1$ be a rational map of degree $q > 1$; or just a quadratic binomial $z^2 + c$, if one prefers. The Julia set $J_f$ is the closure of the repelling periodic poin …

Littlewood's conjecture on simultaneous rational approximation to a pair of real numbers, $$ \liminf_{n \in \mathbb{N}} \, n \cdot \mathrm{dist}(n\alpha,\mathbb{Z}) \cdot \mathrm{dist}(n\beta, \mathbb …

For a fixed $\alpha$, the number $N_{\alpha}(\epsilon)$ is bounded by a polynomial function of $1/\epsilon$. The proof of this requires either Faltings's product theorem, or Esnault and Viehweg's mult …

Let $\mathcal{R}_d \subset \bar{\mathbb{Q}}$ be the set of all roots of degree $d$ polynomials with $\{-1,0,1\}$ coefficients and $$ c(d) := \min_{\substack{ \alpha, \beta \in \mathcal{R}_d \\ \alpha^ …

This result is due originally to K. Mahler, and holds true more generally with any algebraic $a$ having $|a| > 1$ (so that the series converges absolutely). I can recommend Masser's lecture in the CIM …

This is true. Choose any linear subspace $Z$ (over $K$) of dimension $n- d-1$ and disjoint from $X$, and take $\pi : \mathbb{P}_K^n \setminus Z \to \mathbb{P}_K^d$ the corresponding linear projection. …