RECORD
Hausdorff dimension in inhomogeneous Diophantine approximation
Number Theory · Submitted 26 May 2018
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We show that the set of $\epsilon$-badly approximable numbers \[ \mathrm{Bad}^\varepsilon (\alpha) := \{x\in [0,1]\, : \, \liminf_{|q| \to \infty} |q| \cdot \| q\alpha -x \| \geq \varepsilon \} \] has full Hausdorff dimension for some positive $\epsilon$ if and only if $\alpha$ is singular on average.
The compiled record
The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
Open the compiled record
The abstract, sentence by sentence
Let $\alpha$ be an irrational real number. We show that the set of $\epsilon$-badly approximable numbers \[ \mathrm{Bad}^\varepsilon (\alpha) := \{x\in [0,1]\, : \, \liminf_{|q| \to \infty} |q| \cdot \| q\alpha -x \| \geq \varepsilon \} \] has full Hausdorff dimension for some positive $\epsilon$ if and only if $\alpha$ is singular on average. The condition is equivalent to the average $\frac{1}{k} \sum_{i=1, \cdots, k} \log a_i$ of the logarithms of the partial quotients $a_i$ of $\alpha$ going to infinity with $k$. We also consider one-sided approximation, obtain a stronger result when $a_i$ tends to infinity, and establish a partial result in higher dimensions.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Let $\alpha$ be an irrational real number. | background 0.99 | 0.05 | 0.01 | asserted 0.53 |
| s1 | We show that the set of $\epsilon$-badly approximable numbers \[ \mathrm{Bad}^\varepsilon (\alpha) := \{x\in [0,1]\, : \, \liminf_{|q| \to \infty} |q| \cdot \| q\alpha -x \| \geq \varepsilon \} \] has full Hausdorff dimension for some positive $\epsilon$ if and only if $\alpha$ is singular on average. | result 0.99 | 0.95 | 0.93 | asserted 1.00 |
| s2 | The condition is equivalent to the average $\frac{1}{k} \sum_{i=1, \cdots, k} \log a_i$ of the logarithms of the partial quotients $a_i$ of $\alpha$ going to infinity with $k$. | interpretation 0.51 | 0.68 | 0.09 | asserted 1.00 |
| s3 | We also consider one-sided approximation, obtain a stronger result when $a_i$ tends to infinity, and establish a partial result in higher dimensions. | contribution 0.82 | 0.95 | 0.04 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v1
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
The paper
Original abstract
Let $\alpha$ be an irrational real number. We show that the set of $\epsilon$-badly approximable numbers \[ \mathrm{Bad}^\varepsilon (\alpha) := \{x\in [0,1]\, : \, \liminf_{|q| \to \infty} |q| \cdot \| q\alpha -x \| \geq \varepsilon \} \] has full Hausdorff dimension for some positive $\epsilon$ if and only if $\alpha$ is singular on average. The condition is equivalent to the average $\frac{1}{k} \sum_{i=1, \cdots, k} \log a_i$ of the logarithms of the partial quotients $a_i$ of $\alpha$ going to infinity with $k$. We also consider one-sided approximation, obtain a stronger result when $a_i$ tends to infinity, and establish a partial result in higher dimensions.
Human review
How the literature uses it
1 arXiv paper cite this in 6 sentences
Papers citing it · 3 sentences without a confident label
Builds on it 1 paper
-
As a by-product, we obtain results on the dimension of a certain set of inhomogeneous shifts for weighted $$-badly approximable matrices, namely a weighted generalisation of [ref] .
Diophantine transference inequalities: weighted, inhomogeneous, and intermediate exponents · Introduction · 0.88
Uses its method or tool 1 paper
-
In this section, we establish Theorem [ref], closely following [ref] .
Diophantine transference inequalities: weighted, inhomogeneous, and intermediate exponents · Weighted inhomogeneous Bad · 0.89
Cites it as background 1 paper
-
Subsequently, the unweighted sets $^(A)$ were investigated in [ref] , where necessary and sufficient conditions were obtained in dimension $1$, so that $(^(v)) = 1$ for some $> 0$.
Diophantine transference inequalities: weighted, inhomogeneous, and intermediate exponents · Weighted inhomogeneous Bad · 0.89
Discussion
Know something about this paper? Leave a comment, or highlight any passage above to comment on it.