Exact uniform approximation and Dirichlet spectrum in dimension at least two

Abstract

For $m\ge 2$, we determine the Dirichlet spectrum in ${\mathbb {R}}^m$ with respect to simultaneous approximation and the maximum norm as the entire interval [0, 1]. This complements previous work of several authors, especially Akhunzhanov and Moshchevitin, who considered $m=2$ and Euclidean norm. We construct explicit examples of real Liouville vectors realizing any value in the unit interval. In particular, for positive values, they are neither badly approximable nor singular. Thereby we obtain a constructive proof of the main claim in a recent paper by Beresnevich, Guan, Marnat, Ramírez and Velani, who proved existence of such vectors but without being able to provide any concrete value in the Dirichlet spectrum. Our constructive proof is considerably shorter and less involved than previous work on the topic. Moreover, it is flexible enough to show that the according set of vectors with prescribed Dirichlet constant has large packing dimension and rather large Hausdorff dimension as well, thereby contributing to the metrical problem raised in the aforementioned paper by Beresnevich et alia. We further establish a more general result on exact uniform approximation, applicable to a wide class of approximating functions. Moreover, minor twists in the proof yield similar, slightly weaker results when restricting to a certain class of classical fractals or considering other norms on ${\mathbb {R}}^m$. In an Appendix we address the situation of a linear form.

Appendix: The linear form problem

Let $\langle .,.\rangle$ be the standard scalar product on ${\mathbb {R}}^m$ and for $\underline{y}=(y_1,\ldots ,y_m)\in {\mathbb {R}}^m$ denote by $|\underline{y}|_{\infty }= \max _{1\le i\le m} |y_i|$ the maximum norm. Recall further the notation $\Vert .\Vert$ introduced in Sect. 1. For $c^{*}\in (0,1]$, let $\mathrm {Di_m}^{*}(c^{*})$ be the set of $\underline{\xi }\in {\mathbb {R}}^m$ for which the system

$$\begin{aligned} 0<|\underline{y}|_{\infty } \le Q^{*}, \qquad \Vert \langle \underline{y},\underline{\xi }\rangle \Vert \le c^{*}Q^{*-m} \end{aligned}$$

(49)

has a solution in an integer vector $\underline{y}$, for all large parameters $Q^{*}$. By a variant of Dirichlet’s Theorem we have $\mathrm {Di_m}^{*}(1)={\mathbb {R}}^m$ for any $m\ge 1$. Let again $\mathrm {Di_m}^{*}= \cup _{c<1} \mathrm {Di_m}^{*}(c)$. Let $\mathrm {Bad_m}^{*}$ be the set of badly approximable linear forms, its defining property being that for some $c^{*}>0$ and all $Q^{*}$ there is no integer vector solution to (49). For completeness further define accordingly $\mathrm {Sing_m}^{*}= \cap _{c>0} \mathrm {Di_m}^{*}(c)$ and $\varvec{FS}_m^{*}= \mathrm {Di_m}^{*}\setminus (\mathrm {Bad_m}^{*}\cup \mathrm {Sing_m}^{*})$. We point out the well-known identities

$$\begin{aligned} \mathrm {Di_m}=\mathrm {Di_m}^{*} , \qquad \mathrm {Sing_m}=\mathrm {Sing_m}^{*}, \qquad \mathrm {Bad_m}=\mathrm {Bad_m}^{*}, \qquad \varvec{FS}_m=\varvec{FS}_m^{*}. \end{aligned}$$

Note however that the sets $\mathrm {Di_m}(c)$ and $\mathrm {Di_m}^{*}(c)$ do not coincide for the same parameter $c<1$. Nevertheless, it has been shown in [30] appearing after the first version of the present paper that the according Dirichlet spectrum ${\mathbb {D}}_m^{*}$ with respect to one linear form equals [0, 1] as well. Similar metrical results as in Sect. 2, Sect. 3 can be shown for the linear form setting as well, however details are not explicitly carried out in [30]. From Corollary 1 and a transference result phrased below, we obtain the following more modest claim that preceded [30].

Theorem 11.1

For $m\ge 2$ an integer and any $c^{*}\in (0,1]$, if we let

$$\begin{aligned} \omega =\omega (m,c^{*}):= (m+1)^{-m^2-m}(c^{*})^{m^2} \in (0,c^{*}), \end{aligned}$$

then the set

$$\begin{aligned} \mathrm {Di_m}^{*}(c^{*})\setminus (\mathrm {Di_m}^{*}(\omega ) \cup \mathrm {Bad_m}^{*}) \subseteq \varvec{FS}_m^{*} \end{aligned}$$

has packing dimension at least $m-1$ and Hausdorff dimension at least as in Theorem 3.1.

As $m\rightarrow \infty$, the value $\omega (m,c^{*})$ asymptotically satisfies

$$\begin{aligned} \omega (m,c^{*}) > (c^{*})^{m^2}e^{-m^2 \log m-o(m^2 \log m)}. \end{aligned}$$

Our result should be compared with the following partial claim of [4, Theorem 1.5] (see also [25]) obtained from a very different, unconstructive method.

Theorem 11.2

(Beresnevich, Guan, Marnat, Ramírez, Velani) Let $m\ge 2$ an integer and

$$\begin{aligned} \kappa _m = e^{-20(m+1)^3(m+10)}. \end{aligned}$$

Then the set

$$\begin{aligned} \mathrm {Di_m}^{*}(c^{*})\setminus (\mathrm {Di_m}^{*}(\kappa _mc^{*}) \cup \mathrm {Bad_m}) \subseteq \varvec{FS}_m^{*} \end{aligned}$$

is uncountable.

We should remark that in the statement we suppressed some more information on other exponents of approximation given in [4, Theorem 1.5]. Moreover, the exact shape of the polynomial in the exponent of $\kappa _m$, in particular the leading coefficient 20, can be readily optimized with sharper estimates at certain places in [4]. Note that in contrast to our result, the value $\kappa _m$ in Theorem 11.2 is independent from $c^{*}$. We see that for large m and large enough $c^{*}\in (0,1]$, roughly as soon as $c^{*} > e^{-m^2}$, our bound from Theorem 11.1 is stronger. For $c^{*}$ very close to 1, we basically can reduce the quartic polynomial in m within the exponent in $\kappa _m$ to a quadratic polynomial.

For the deduction of Theorem 11.1 it is convenient to apply a transference result by German [14] based on geometry of numbers, which improves on previous results by Mahler. Concretely, we use the following special cases of [14, Theorem 7], where we implicitly include the upper estimate $\Delta _d^{-1}\le d^{1/2}$ from [14, § 2] for the quantity $\Delta _d$ defined there, where $d=m+1$ in our situation.

Theorem 11.3

(German) Let $m\ge 1$ and $\underline{\xi }\in {\mathbb {R}}^m$. Let X, U be positive parameters.

• (i) Let $x\in {\mathbb {Z}}$ and assume

  $$\begin{aligned} 0<|x|\le X, \qquad \Vert \underline{\xi }x\Vert \le U. \end{aligned}$$

  Then there exists $\underline{y}^{*}\in {\mathbb {Z}}^m$ so that

  $$\begin{aligned} 0< |\underline{y}^{*}|_{\infty }\le Y, \qquad \Vert \langle \underline{y}^{*},\underline{\xi }\rangle \Vert \le V, \end{aligned}$$

  where

  $$\begin{aligned} Y= (m+1)^{1/(2m)}X^{1/m}, \qquad V= (m+1)^{1/(2m)}X^{1/m-1}U. \end{aligned}$$

• (ii) Let $\underline{y}\in {\mathbb {Z}}^m$ and assume

  $$\begin{aligned} 0<|\underline{y}|_{\infty }\le X, \qquad \Vert \langle \underline{y},\underline{\xi }\rangle \Vert \le U. \end{aligned}$$

  Then there exists $x^{\prime }\in {\mathbb {Z}}$ so that

  $$\begin{aligned} 0< |x^{\prime }|\le Y^{\prime }, \qquad \Vert x^{\prime }\underline{\xi }\Vert \le V^{\prime } \end{aligned}$$

  where

  $$\begin{aligned} Y^{\prime }= (m+1)^{1/(2m)}XU^{1/m-1}, \qquad V^{\prime }= (m+1)^{1/(2m)}U^{1/m}. \end{aligned}$$

We now prove our claim.

Proof of Theorem 11.1

Let $c\in (0,1]$ to be chosen later. By Corollary 1 and Theorem 3.1, the set $\mathrm {Di_m}(c)\setminus (\cup _{\epsilon>0} \mathrm {Di_m}(c-\epsilon ) \cup \mathrm {Bad_m})=\mathrm {Di_m}(c)\setminus (\cup _{\epsilon >0} \mathrm {Di_m}(c-\epsilon ) \cup \mathrm {Bad_m}^{*})$ has the stated metrical properties. Take any $\underline{\xi }$ in this set. Take arbitrary, large Y and put $X=(Y(m+1)^{-1/(2m)})^m$, which is also large. Now the hypothesis of (i) from Theorem 11.3 holds when we take

$$\begin{aligned} U=cX^{-1/m}. \end{aligned}$$

From the conclusion we get $\underline{y}^{*}\in {\mathbb {Z}}^m$ that satisfies

$$\begin{aligned} 0< |\underline{y}^{*}|_{\infty }\le Y=(m+1)^{1/(2m)}X^{1/m} \end{aligned}$$

and

$$\begin{aligned} \Vert \langle \underline{y}^{*},\underline{\xi }\rangle \Vert \le V= (m+1)^{1/(2m)}X^{1/m-1}U= c(m+1)^{1/(2m)}X^{-1}= c^{*}Y^{-m}, \end{aligned}$$

where

$$\begin{aligned} c^{*}= c(m+1)^{1/2+1/(2m)}. \end{aligned}$$

(50)

Observe this holds for all large Y, so $\underline{\xi }\in \mathrm {Di_m}^{*}(c^{*})$.

Now assume that for some $\tilde{c}\in (0,1)$ we have $\underline{\xi }\in {\textrm{Di}_\textrm{m}}^{*}(\tilde{c})$. That means for all large X we may take

$$\begin{aligned} U= \tilde{c} X^{-m} \end{aligned}$$

and the hypothesis of (ii) from Theorem 11.3 is satisfied for some $\underline{y}\in {\mathbb {Z}}^m$. From the conclusion we get the existence of a positive integer $x^{\prime }$ satisfying

$$\begin{aligned} 0< |x^{\prime }|\le Y^{\prime }= (m+1)^{1/(2m)}XU^{1/m-1}= (m+1)^{1/(2m)} \tilde{c}^{1/m-1} X^m \end{aligned}$$

and

$$\begin{aligned} \Vert x^{\prime }\underline{\xi }\Vert \le V^{\prime }= (m+1)^{1/(2m)}U^{1/m}=(m+1)^{1/(2m)} \tilde{c}^{1/m} X^{-1}= \tilde{C}Y^{\prime -1/m}, \end{aligned}$$

where

$$\begin{aligned} \tilde{C}= (m+1)^{1/2+1/(2m)} \tilde{c}^{1/m^2}. \end{aligned}$$

(51)

Now, since $Y^{\prime }$ can be any large number by choosing X suitably, we infer $\underline{\xi }\in \mathrm {Di_m}$ $(\tilde{C})$. If $\tilde{C}< c$ and assuming $c\le 1$, this contradicts our choice of $\underline{\xi }$. By (50), the latter condition $c\le 1$ clearly holds as soon as $c^{*}\le 1$, so we require $\tilde{C}\ge c$, which by (51) and (50) leads to

$$\begin{aligned} \tilde{c}\ge ((m+1)^{-1/2-1/(2m)}c)^{m^2}= (m+1)^{-m^2/2-m/2} c^{m^2}= (m+1)^{-m^2-m}c^{*m^2}. \end{aligned}$$

Combining our results we see that $\underline{\xi }\in \mathrm {Di_m}^{*}(c^{*})\setminus (\mathrm {Di_m}^{*}((m+1)^{-m^2-m}c^{*m^2})\cup \mathrm {Bad_m}^{*})$. $\square$

It may be possible to derive similar, possibly stronger, effective results when combining Corollary 1 with the essential method of Davenport and Schmidt [10], however in reverse direction (we need the conclusion from simultaneous approximation to linear form instead of the other way round), instead of [14]. We also want to refer to [4, § 4], in particular [4, Lemma 4.9], in this context.