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Given $x \in \mathbb{R}$ we will write $\{x\}$ for the fractional part of $x$ and $\|x\|$ for the distance of $x$ from the nearest integer, in such a way that $\{x\} = x - \lfloor x \rfloor$ and $\|x\| = \min(\{x\}, 1 - \{x\})$, where $\lfloor x \rfloor$ is, as usual, the greatest integer $\le x$.

Let $x$ be a fixed irrational number and $\mu$ its irrationality measure, namely the infimum of the set of all positive exponents $s$ such that

$$0 < \left|x - \frac{m}{n}\right| < \frac{1}{n^s}$$ for finitely many pairs $(m,n) \in \mathbb{Z} \times \mathbb{N}^+$. Recall that $\mu \ge 2$ by (a corollary of) Dirichlet's (approximation) theorem. Then, given $s \in \{2\} \cup [2,\mu[$, define $G_s$ as the set

$$\{n \in \mathbb{N}^+: \|nx\| < n^{1-s}\},$$ and let $G_s^{-} := \big\{n \in G_s: \|nx\| = \{nx\}\big\}$ and $G_s^+ := G_s \setminus G_s^{-}$.

We have that $G_s$ is infinite (by Dirichlet's theorem and the definition of $\mu$), so at least one of $G_s^-$ or $G_s^+$ is infinite too, and I'm tempted to claim that each of them must be infinite. I don't have a serious argument in support of this: My only point is that it would be weird, I believe, to observe a similar "asymmetry" in the diophantine approximations of $x$ (yet, I would be happy to hear that my expectation is completely wrong). This leads to the following:

Q1. Is it true that $G_s^-$ is infinite for each $s \in \{2\} \cup [2,\mu[$? Q2. If Q1 is well-established, would you kindly provide me with a reference?

Some remarks: (i) If the answer to Q1 is yes, then it is as well true that $G_s^+$ is infinite for each $s \in \{2\} \cup [2,\mu[$. (ii) I can prove that the answer to Q1 is positive if $s = 2$ (this follows from Dirichlet's theorem and some elementary properties of the simple continued fraction expansion of $x$). (iii) By (ii) and Khintchine's theorem (which yields that the irrationality measure of most real numbers is equal to $2$), the answer to Q1 is yes for almost all $x$.

Given $x \in \mathbb{R}$ we will write $\{x\}$ for the fractional part of $x$ and $\|x\|$ for the distance of $x$ from the nearest integer, in such a way that $\{x\} = x - \lfloor x \rfloor$ and $\|x\| = \min(\{x\}, 1 - \{x\})$, where $\lfloor x \rfloor$ is, as usual, the greatest integer $\le x$.

Let $x$ be a fixed irrational number and $\mu$ its irrationality measure, namely the infimum of the set of all positive exponents $s$ such that

$$0 < \left|x - \frac{m}{n}\right| < \frac{1}{n^s}$$ for finitely many pairs $(m,n) \in \mathbb{Z} \times \mathbb{N}^+$. Assume that $\mu$ is finite (added later: see N. Elkies' answer and the comments below). Recall that $\mu \ge 2$ by (a corollary of) Dirichlet's (approximation) theorem. Then, given $s \in \{2\} \cup [2,\mu[$, define $G_s$ as the set

$$\{n \in \mathbb{N}^+: \|nx\| < n^{1-s}\},$$ and let $G_s^{-} := \big\{n \in G_s: \|nx\| = \{nx\}\big\}$ and $G_s^+ := G_s \setminus G_s^{-}$.

We have that $G_s$ is infinite (by Dirichlet's theorem and the definition of $\mu$), so at least one of $G_s^-$ or $G_s^+$ is infinite too, and I'm tempted to claim that each of them must be infinite. I don't have a serious argument in support of this: My only point is that it would be weird, I believe, to observe a similar "asymmetry" in the diophantine approximations of $x$ (yet, I would be happy to hear that my expectation is completely wrong). This leads to the following:

Q1. Is it true that $G_s^-$ is infinite for each $s \in \{2\} \cup [2,\mu[$? Q2. If Q1 is well-established, would you kindly provide me with a reference?

Some remarks: (i) If the answer to Q1 is yes, then it is as well true that $G_s^+$ is infinite for each $s \in \{2\} \cup [2,\mu[$. (ii) I can prove that the answer to Q1 is positive if $s = 2$ (this follows from Dirichlet's theorem and some elementary properties of the simple continued fraction expansion of $x$). (iii) By (ii) and Khintchine's theorem (which yields that the irrationality measure of most real numbers is equal to $2$), the answer to Q1 is yes for almost all $x$.

Given $x \in \mathbb{R}$ we will write $\{x\}$ for the fractional part of $x$ and $\|x\|$ for the distance of $x$ from the nearest integer, in such a way that $\{x\} = x - \lfloor x \rfloor$ and $\|x\| = \min(\{x\}, 1 - \{x\})$, where $\lfloor x \rfloor$ is, as usual, the greatest integer $\le x$.

Let $x$ be a fixed irrational number and $\mu$ its irrationality measure, namely the infimum of the set of all positive exponents $s$ such that

$$0 < \left|x - \frac{m}{n}\right| < \frac{1}{n^s}$$ for finitely many pairs $(m,n) \in \mathbb{Z} \times \mathbb{N}^+$. Recall that $\mu \ge 2$ by (a corollary of) Dirichlet's (approximation) theorem. Then, given $s \in \{2\} \cup [2,\mu[$, define $G_s$ as the set

$$\{n \in \mathbb{N}^+: \|nx\| < n^{1-s}\},$$ and let $G_s^{-} := \big\{n \in G_s: \|nx\| = \{nx\}\big\}$ and $G_s^+ := G_s \setminus G_s^{-}$.

We have that $G_s$ is infinite (by Dirichlet's theorem and the definition of $\mu$), so at least one of $G_s^-$ or $G_s^+$ is infinite too, and I'm tempted to claim that each of them must be infinite. I don't have a serious argument in support of this: My only point is that it would be weird, I believe, to observe a similar "asymmetry" in the diophantine approximations of $x$ (yet, I would be happy to hear that my expectation is completely wrong). This leads to the following:

Q. (1) Is it true that $G_s^-$ is infinite for each $s \in \{2\} \cup [2,\mu[$? (2) And, if this is well-established, would you kindly provide me with a reference?

A couple of remarks: (1) If the answer to Q(1) is yes, then it is as well true that $G_s^+$ is infinite for each $s \in \{2\} \cup [2,\mu[$. (2) I can prove that the answer to Q1 is positive if $s = 2$ (this follows from Dirichlet's theorem and some elementary properties of the simple continued fraction expansion of $x$).

Given $x \in \mathbb{R}$ we will write $\{x\}$ for the fractional part of $x$ and $\|x\|$ for the distance of $x$ from the nearest integer, in such a way that $\{x\} = x - \lfloor x \rfloor$ and $\|x\| = \min(\{x\}, 1 - \{x\})$, where $\lfloor x \rfloor$ is, as usual, the greatest integer $\le x$.

Let $x$ be a fixed irrational number and $\mu$ its irrationality measure, namely the infimum of the set of all positive exponents $s$ such that

$$0 < \left|x - \frac{m}{n}\right| < \frac{1}{n^s}$$ for finitely many pairs $(m,n) \in \mathbb{Z} \times \mathbb{N}^+$. Recall that $\mu \ge 2$ by (a corollary of) Dirichlet's (approximation) theorem. Then, given $s \in \{2\} \cup [2,\mu[$, define $G_s$ as the set

$$\{n \in \mathbb{N}^+: \|nx\| < n^{1-s}\},$$ and let $G_s^{-} := \big\{n \in G_s: \|nx\| = \{nx\}\big\}$ and $G_s^+ := G_s \setminus G_s^{-}$.

We have that $G_s$ is infinite (by Dirichlet's theorem and the definition of $\mu$), so at least one of $G_s^-$ or $G_s^+$ is infinite too, and I'm tempted to claim that each of them must be infinite. I don't have a serious argument in support of this: My only point is that it would be weird, I believe, to observe a similar "asymmetry" in the diophantine approximations of $x$ (yet, I would be happy to hear that my expectation is completely wrong). This leads to the following:

Q1. Is it true that $G_s^-$ is infinite for each $s \in \{2\} \cup [2,\mu[$? Q2. If Q1 is well-established, would you kindly provide me with a reference?

Some remarks: (i) If the answer to Q1 is yes, then it is as well true that $G_s^+$ is infinite for each $s \in \{2\} \cup [2,\mu[$. (ii) I can prove that the answer to Q1 is positive if $s = 2$ (this follows from Dirichlet's theorem and some elementary properties of the simple continued fraction expansion of $x$). (iii) By (ii) and Khintchine's theorem (which yields that the irrationality measure of most real numbers is equal to $2$), the answer to Q1 is yes for almost all $x$.

Given $x \in \mathbb{R}$ we will write $\{x\}$ for the fractional part of $x$ and $\|x\|$ for the distance of $x$ from the nearest integer, in such a way that $\{x\} = x - \lfloor x \rfloor$ and $\|x\| = \min(\{x\}, 1 - \{x\})$, where $\lfloor x \rfloor$ is, as usual, the greatest integer $\le x$.

Let $x$ be a fixed irrational number and $\mu$ its irrationality measure, namely the infimum of the set of all positive exponents $s$ such that

$$0 < \left|x - \frac{m}{n}\right| < \frac{1}{n^s}$$ for finitely many pairs $(m,n) \in \mathbb{Z} \times \mathbb{N}^+$. Recall that $\mu \ge 2$ by (a corollary of) Dirichlet's (approximation) theorem. Then, given $s \in \{2\} \cup [2,\mu[$, define $G_s$ as the set

$$\{n \in \mathbb{N}^+: \|nx\| < n^{1-s}\},$$ and let $G_s^{-} := \big\{n \in G_s: \|nx\| = \{nx\}\big\}$ and $G_s^+ := G_s \setminus G_s^{-}$.

We have that $G_s$ is infinite (by Dirichlet's theorem and the definition of $\mu$), so at least one of $G_s^-$ or $G_s^+$ is infinite too, and I'm tempted to claim that each of them must be infinite. I don't have a serious argument in support of this: My only point is that it would be weird, I believe, to observe a similar "asymmetry" in the diophantine approximations of $x$ (yet, I would be happy to hear that my expectation is completely wrong). This leads to the following:

Q. (1) Is it true that $G_s^-$ is infinite for each $s \in \{2\} \cup [2,\mu[$? (2) And, if this is well-established, would you kindly provide me with a reference?

A couple of remarks: (1) If the answer to Q(1) is yes, then it is as well true that $G_s^+$ is infinite for each $s \in \{2\} \cup [2,\mu[$. (2) I can prove that the answer to Q1 is positive if $s = 2$ (this follows from elementary properties of the expansion of $x$ in a simple continued fraction).

Given $x \in \mathbb{R}$ we will write $\{x\}$ for the fractional part of $x$ and $\|x\|$ for the distance of $x$ from the nearest integer, in such a way that $\{x\} = x - \lfloor x \rfloor$ and $\|x\| = \min(\{x\}, 1 - \{x\})$, where $\lfloor x \rfloor$ is, as usual, the greatest integer $\le x$.

Let $x$ be a fixed irrational number and $\mu$ its irrationality measure, namely the infimum of the set of all positive exponents $s$ such that

$$0 < \left|x - \frac{m}{n}\right| < \frac{1}{n^s}$$ for finitely many pairs $(m,n) \in \mathbb{Z} \times \mathbb{N}^+$. Recall that $\mu \ge 2$ by (a corollary of) Dirichlet's (approximation) theorem. Then, given $s \in \{2\} \cup [2,\mu[$, define $G_s$ as the set

$$\{n \in \mathbb{N}^+: \|nx\| < n^{1-s}\},$$ and let $G_s^{-} := \big\{n \in G_s: \|nx\| = \{nx\}\big\}$ and $G_s^+ := G_s \setminus G_s^{-}$.

We have that $G_s$ is infinite (by Dirichlet's theorem and the definition of $\mu$), so at least one of $G_s^-$ or $G_s^+$ is infinite too, and I'm tempted to claim that each of them must be infinite. I don't have a serious argument in support of this: My only point is that it would be weird, I believe, to observe a similar "asymmetry" in the diophantine approximations of $x$ (yet, I would be happy to hear that my expectation is completely wrong). This leads to the following:

Q. (1) Is it true that $G_s^-$ is infinite for each $s \in \{2\} \cup [2,\mu[$? (2) And, if this is well-established, would you kindly provide me with a reference?

A couple of remarks: (1) If the answer to Q(1) is yes, then it is as well true that $G_s^+$ is infinite for each $s \in \{2\} \cup [2,\mu[$. (2) I can prove that the answer to Q1 is positive if $s = 2$ (this follows from Dirichlet's theorem and some elementary properties of the simple continued fraction expansion of $x$).