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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, 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 (in $\deg{\beta} \to \infty$) approximation by such a sequence, meaning $ -\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 each of these notions of arithmetic smallness:

(i). All algebraic points $\alpha \in \mathbb{G}_m(\bar{\mathbb{Q}})$ with $|\alpha| \neq 1$ attract arithmetically small points. Replacing $\alpha$ with $1/\alpha$ it is enough to consider $|\alpha| > 1$. Then, as $n \to \infty$, the solution near $\alpha$ of $X^n(X-\alpha) = 1$ satisfies (a) and approaches $\alpha$ exponentially fast with $c(\alpha) = \log{|\alpha|}- \epsilon$. Since this $c(\alpha)$ is continuously dependent on $|\alpha|$, we see moreover by an approximation with algebraic points and an extraction of a diagonal limit that, with the notions (b) and (c), all $\alpha \in \mathbb{\mathbb{C}}$ with $|\alpha| \neq 1$ attract arithmetically small points.

(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 $|\alpha| = 1$, other than the roots of unity? 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 that are not roots of unity?

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, 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 (in $\deg{\beta} \to \infty$) approximation by such a sequence, meaning $ -\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 each of these notions of arithmetic smallness:

(i). All algebraic points $\alpha \in \mathbb{G}_m(\bar{\mathbb{Q}})$ with $|\alpha| \neq 1$ attract arithmetically small points. Replacing $\alpha$ with $1/\alpha$ it is enough to consider $|\alpha| > 1$. Then, as $n \to \infty$, the solution near $\alpha$ of $X^n(X-\alpha) = 1$ satisfies (a) and approaches $\alpha$ exponentially fast with $c(\alpha) = \log{|\alpha|}- \epsilon$. Since this $c(\alpha)$ is continuously dependent on $|\alpha|$, we see moreover by an approximation with algebraic points and an extraction of a diagonal limit that, with the notions (b) and (c), all $\alpha \in \mathbb{G}_m(\mathbb{\mathbb{C}})$ with $|\alpha| \neq 1$ attract arithmetically small points.

(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 $|\alpha| = 1$, other than the roots of unity? 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 that are not roots of unity?

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, 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 (in $\deg{\beta} \to \infty$) approximation by such a sequence, meaning $ -\log{|\alpha - \beta|} > c(\alpha) \cdot \deg{\beta}$, where $c(\alpha) > 0$ and depends only on $\alpha$.

Which $\alpha$ have this property, for any of these notions of arithmetic smallness? The following is easy to see:

(i). All algebraic points $\alpha \in \mathbb{G}_m(\bar{\mathbb{Q}})$ with $|\alpha| \neq 1$ attract arithmetically small points (in either variant). Replacing $\alpha$ with $1/\alpha$ it is enough to consider $|\alpha| > 1$. Then, as $n \to \infty$, the solution near $\alpha$ of $X^n(X-\alpha) = 1$ satisfies (a) and approaches $\alpha$ exponentially fast with $c(\alpha) = \log{|\alpha|}- \epsilon$.

(ii) All complex points $\alpha \in \mathbb{G}_m(\mathbb{C})$ with $|\alpha| \neq 1$ attract arithmetically small points in the variant (b) (and hence also in (c)). For, since the $c(\alpha)$ constructed in (i) is continuously dependent on $|\alpha|$, we may approximate $\alpha$ by Gaussian rationals and extract a diagonal limit.

(iii). Roots of unity do not attract arithmetically small points (in either variant). 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.

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

(v). 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 $|\alpha| = 1$ not roots of unity? 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?

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, 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 (in $\deg{\beta} \to \infty$) approximation by such a sequence, meaning $ -\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 each of these notions of arithmetic smallness:

(i). All algebraic points $\alpha \in \mathbb{G}_m(\bar{\mathbb{Q}})$ with $|\alpha| \neq 1$ attract arithmetically small points. Replacing $\alpha$ with $1/\alpha$ it is enough to consider $|\alpha| > 1$. Then, as $n \to \infty$, the solution near $\alpha$ of $X^n(X-\alpha) = 1$ satisfies (a) and approaches $\alpha$ exponentially fast with $c(\alpha) = \log{|\alpha|}- \epsilon$. Since this $c(\alpha)$ is continuously dependent on $|\alpha|$, we see moreover by an approximation with algebraic points and an extraction of a diagonal limit that, with the notions (b) and (c), all $\alpha \in \mathbb{\mathbb{C}}$ with $|\alpha| \neq 1$ attract arithmetically small points.

(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 $|\alpha| = 1$, other than the roots of unity? 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 that are not roots of unity?

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, 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 (in $\deg{\beta} \to \infty$) approximation by such a sequence, meaning $ -\log{|\alpha - \beta|} > c(\alpha) \cdot \deg{\beta}$, where $c(\alpha) > 0$ and depends only on $\alpha$.

Which $\alpha$ have this property, for each of these notions of arithmetic smallness? The following is easy to see:

(i). All algebraic points $\alpha \in \mathbb{G}_m(\bar{\mathbb{Q}})$ with $|\alpha| \neq 1$ attract arithmetically small points in the notion (a) (and hence also in (b) and (c)). Replacing $\alpha$ with $1/\alpha$ it is enough to consider $|\alpha| > 1$. Then, as $n \to \infty$, the solution near $\alpha$ of $X^n(X-\alpha) = 1$ satisfies (a) and approaches $\alpha$ exponentially fast with $c(\alpha) = \log{|\alpha|}- \epsilon$.

(ii) All complex points $\alpha \in \mathbb{G}_m(\mathbb{C})$ with $|\alpha| \neq 1$ attract arithmetically small points in the notion (b) (and hence also in (c)). For, since the $c(\alpha)$ constructed in (i) is continuously dependent on $|\alpha|$, we may approximate $\alpha$ by Gaussian rationals and extract a diagonal limit.

(iii). Roots of unity do not attract arithmetically small points, in the notion (c) (and hence also in (a) and (b)). 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.

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

(v). 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 $|\alpha| = 1$ not roots of unity? 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?

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, 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 (in $\deg{\beta} \to \infty$) approximation by such a sequence, meaning $ -\log{|\alpha - \beta|} > c(\alpha) \cdot \deg{\beta}$, where $c(\alpha) > 0$ and depends only on $\alpha$.

Which $\alpha$ have this property, for any of these notions of arithmetic smallness? The following is easy to see:

(i). All algebraic points $\alpha \in \mathbb{G}_m(\bar{\mathbb{Q}})$ with $|\alpha| \neq 1$ attract arithmetically small points (in either variant). Replacing $\alpha$ with $1/\alpha$ it is enough to consider $|\alpha| > 1$. Then, as $n \to \infty$, the solution near $\alpha$ of $X^n(X-\alpha) = 1$ satisfies (a) and approaches $\alpha$ exponentially fast with $c(\alpha) = \log{|\alpha|}- \epsilon$.

(ii) All complex points $\alpha \in \mathbb{G}_m(\mathbb{C})$ with $|\alpha| \neq 1$ attract arithmetically small points in the variant (b) (and hence also in (c)). For, since the $c(\alpha)$ constructed in (i) is continuously dependent on $|\alpha|$, we may approximate $\alpha$ by Gaussian rationals and extract a diagonal limit.

(iii). Roots of unity do not attract arithmetically small points (in either variant). 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.

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

(v). 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 $|\alpha| = 1$ not roots of unity? 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?