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 algebraic points $\beta$ that, in order of decreasing restrictiveness, are either:

(a) Roots of integer polynomials of bounded *lengths* (sum of absolute values of all coefficients);

(b) Roots of $f = \sum a_iX^i \in \mathbb{Z}[X]$, $f(0) \neq 0$, of subexponental length: $\log(\sum |a_i|) = o(\deg{f})$;

(c) Roots of $f = a \prod (X-\beta_i) \in \mathbb{Z}[X]$, $f(0) \neq 0$, having absolute logarithmic height $\frac{1}{\deg{f}}\big( \log{|a|} + \sum \log^+{|\beta_i|} \big) \to 0$.

Let me say that $\alpha \in \mathbb{C}^{\times}$ *attracts arithmetically small points* if it admits an exponentially good approximation by a sequence of such $\beta$ with $\deg{\beta} \to \infty$. That is to say: $ -\log{|\alpha - \beta|} > c(\alpha) \cdot \deg{\beta}$, where $c(\alpha) > 0$ and depends only on $\alpha$.

Which $\alpha$ have this property? The following is easy to see, for all three versions (a), (b) and (c) of arithmetic smallness:

(i). *All points with $|\alpha| \neq 1$ attract arithmetically small points.* Replacing $\alpha$ with $1/\alpha$ it is enough to consider $|\alpha| > 1$. If $\alpha \in \bar{\mathbb{Q}}$, then as $n \to \infty$, the solution nearest to $\alpha$  of $X^n(X-\alpha) = 1$ satisfies (a) and approaches $\alpha$ exponentially fast with $c(\alpha) = |\alpha|^{1/\deg{\alpha}} - \epsilon$. In general, since for $\alpha \in \mathbb{Q}(i)$ this $c(\alpha)$ is continuously dependent on $|\alpha|$, we may approximate $\alpha$ by Gaussian rationals and extract a diagonal limit.

(ii). *Roots of unity do not attract arithmetically small points.* This is immediate from the existence of non-zero integer polynomial multiples of $(X-1)^m$ whose length is subexponential in $m$. Such polynomials, arising as an easy application of Siegel's lemma, have been found and exploited by Mignotte. The best construction is in Bombieri and Vaaler's paper *Polynomials with low height and prescribed vanishing.*

(iii). *No algebraic $\alpha$ attracts roots of unity.* This follows by the Gel'fond-Baker theorem.

(iv). *There are, of course, transcendental values $\alpha \in S^1$ that attract roots of unity.* Exponentiate Liouville's example $\sum 2^{-n!}$. *But those values have zero measure on the circle* (by Khinchin's theorem).

The **questions**, then, are obvious:

1. What happens for algebraic $\alpha$ with $|\alpha| = 1$? Does any such point attract arithmetically small points? 

2. And do the points in $S^1 = \{|z| = 1\}$ that attract arithmetically small points have measure zero on the circle? Indeed, can we construct a point on the circle that attracts arithmetically small points but not roots of unity?