On absolutely normal numbers and their discrepancy estimate

TL;DR

This paper constructs a specific absolutely normal number in base 2 whose digit sequence exhibits discrepancy properties similar to almost all real numbers across all integer bases, advancing understanding of normality and distribution.

Contribution

It provides an explicit construction of an absolutely normal number with discrepancy behavior matching that of typical real numbers across all bases.

Findings

Constructed an explicit absolutely normal number in base 2.

Demonstrated the discrepancy of the sequence matches typical behavior.

Showed uniform distribution properties hold across all integer bases.

Abstract

We construct the base $2$ expansion of an absolutely normal real number $x$ so that, for every integer $b$ greater than or equal to $2$, the discrepancy modulo $1$ of the sequence $(b^0 x, b^1 x, b^2 x , \ldots)$ is essentially the same as that realized by almost all real numbers.

Equations

$$D_N((x_j{)}_{j\ge 1})=\underset{0\le u<v\le 1}{\sup }\left|\frac{\#\{j:1\le j\le N\text{ and }u\le x_j<v\}}{N}-(v-u)\right|.$$

$$\underset{N\to \infty }{\mathrm{limsup}}\frac{D_N(\{b^{j}x{\}}_{j\ge 0})\sqrt{N}}{\sqrt{\log \log N}}<3C_b,$$

$$C_b=166+664/(\sqrt{b}-1)\text{ is Philipp’s constant}.$$

$$\underset{N\to \infty }{\mathrm{limsup}}\frac{D_N(\{2^{j}x{\}}_{j\ge 0})\sqrt{N}}{\sqrt{\log \log N}}<C.$$

$$\underset{N\to \infty }{\mathrm{limsup}}\frac{D_N(\{n_{j}x{\}}_{j\ge 1})\sqrt{N}}{\sqrt{\log \log N}}<166+664/(\sqrt{\theta }-1).$$

$$\underset{N\to \infty }{\mathrm{limsup}}\frac{D_N(\{{\theta }^{j}x{\}}_{j\ge 0})\sqrt{N}}{\sqrt{\log \log N}}=C_{\theta }^{\prime }.$$

$$C^{\prime}_{\theta}=\left\{\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus 2.0pt\sqrt{84}/9,&\text{ if }\theta=2\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\sqrt{2(\theta+1)/(\theta-1)}/2,&\text{ if $\theta$ is odd}\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\sqrt{2(\theta+1)\theta(\theta-2)/(\theta-1)^{3}}/2,&\text{ if }\theta\geq 4\text{ is even}.\end{array}\right.$$

$$D_N(\{{\theta }^{j}x{\}}_{j\ge 0})<C_{\theta }^{\prime \prime }\frac{(\log N{)}^3}{\sqrt{N}}.$$

$$\displaystyle F(M,N,\alpha_{1},\alpha_{2},(x_{j})_{j\geq 1})=\big{|}$$

$$\mu \{x\in (0,1):|F(M,N,a{b}^{-m},(a+1)b^{-m},\{b^{j}x{\}}_{j\ge 0})|>\epsilon N\}$$

$$\mu \left\{x\in (0,1):F(M,N,a{2}^{-k},(a+1)2^{-k},\{b^{j}x{\}}_{j\ge 0})\ge \epsilon N\right\}$$

$$|F(0,N,{\alpha }_1,{\alpha }_2,(x_j{)}_{j\ge 1})|\le N/2^{k-1}+\sum _{m=1}^k\underset{0\le a<2^m}{\max }|F(0,N,a{2}^{-m},(a+1)2^{-m},(x_j{)}_{j\ge 1})|.$$

$$F(0,N,a{2}^{-h},(a+1)2^{-h},\{b^{j}x{\}}_{j\ge 0})$$

$$+$$

$${\tilde{C}}_b$$

$$\varphi (N)$$

$$T(N)$$

$$b\ge 2,n\ge 1,0\le a<2^{T(2^n)},1\le h\le T(2^n),n/2\le \ell \le n,\text{ and }1\le m\le 2^{n/2},$$

$$G(b,n,a,h)=$$

$${\alpha }_1=a{2}^{-(h+1)},{\alpha }_2=(a+1)2^{-(h+1)},\text{ if }1\le h<T(2^n);$$

$${\alpha }_1=a{2}^{-T(2^n)},{\alpha }_2=(a+1)2^{-T(2^n)},\text{ if }h=T(2^n).$$

$$H(b,n,a,h,\ell ,m)=$$

$${\beta }_1=a{2}^{-(h+1)},{\beta }_2=(a+1)2^{-(h+1)},\text{ if }1\le h<T(2^{\ell -1});$$

$${\beta }_1=a{2}^{-T(2^{\ell -1})},{\beta }_2=(a+1)2^{-T(2^{\ell -1})},\text{ if }h=T(2^{\ell -1}).$$

$$G_{b,n}=$$

$$H_{b,n}=$$

$$\mu (G_{b,n})=n^{-1-4\delta },\mu (H_{b,n})=2n^{-1-3\delta },$$

$$\mu \left(\bigcup _{n\ge n_0}(G_{b,n}\cup H_{b,n})\right)<\eta$$

$$\underset{N\to \infty }{\mathrm{limsup}}\frac{D_N(\{b^{n}x{\}}_{n\ge 0})\sqrt{N}}{\sqrt{\log \log N}}<(1+4\delta )C_b,$$

$$\mu G_{b,n}\le$$

$$\le$$

$$+2^{T}9\cdot 2^{2(T+2)}(T+2)\exp \left(-2^{-T/4}{\varphi }^2(2^n)2^{-n}{b}^T\frac{b^2}{6(T+2)}\right)$$

$$\le$$

$$\mu$$

$$\le$$

$$+\sum _{\ell =n/2}^n{2}^{n-\ell }92^{3T+4}(T+2)\exp \left(-2^{-T/4}{b}^{T+2}{2}^{2(n-\ell )/3}\log \log (2^n)(1+\delta {)}^2\frac{4}{6(T+2)}{\tilde{C}}_b^2\right)$$

$$\le$$

$$+\sum _{\ell =n/2}^n{2}^{n-\ell }92^{3T+4}(T+2)\exp \left(-2^{-T/4}{b}^{T+2}{2}^{2(n-\ell )/3}\log \log (2^n)(1+\delta {)}^2\frac{1}{6(T+2)}\right)$$

Full text

This paper has been superseded by:

“On the construction of absolutely normal numbers”

Christoph Aistleitner, Verónica Becher, Adrian-Maria Scheerer and Theodore Slaman

July 2017, arXiv:1707.02628

http://arxiv.org/abs/1707.02628

For a real number $x$, we write $\{x\}=x-\lfloor x\rfloor$ to denote the fractional part of $x$. For a sequence $(x_{j})_{j\geq 1}$ of real numbers in the unit interval, the discrepancy of the $N$ first elements is

[TABLE]

In this note we prove the following.

[TABLE]

where

[TABLE]

The algorithm computes the first $n$ digits of the expansion of $x$ in base $2$ after performing triple-exponential in $n$ mathematical operations.

It is well known that for almost all real numbers $x$ and for all integers $b$ greater than or equal to $2$, the sequence $\{b^{j}x\}_{j\geq 0}$ is uniformly distributed in the unit interval, which means that its discrepancy tends to [math] as $N$ goes to infinity. In [6], Gál and Gál proved that there is a constant $C$ such that for almost all real numbers $x$,

[TABLE]

Philipp [9] bounded the existential constant $C$ and extended this result for lacunary sequences. He proved that given a sequence of positive integers $(n_{j})_{j\geq 1}$ such that $n_{j+1}/n_{j}\geq\theta$ for some real number $\theta>1$, then for almost all real numbers $x$ the sequence $\{n_{j}x\}_{j\geq 1}$ satisfies

[TABLE]

Finally, Fukuyama [5] explicitly determined, for any real $\theta>1$, the constant $C^{\prime}_{\theta}$ (see [5, Corollary]) such that for almost all real numbers $x$,

[TABLE]

For instance, in case $\theta$ is an integer greater than or equal to $2$,

[TABLE]

The proof of Theorem 1 is based on the explicit construction of a set of full Lebesgue measure given by Philipp in [9], which, in turn, follows from that in [6]. Unfortunately we do not know an explicit construction of a set with full Lebesgue measure achieving the constants proved by Fukuyama [5]. If one could give such an explicit construction one could obtain a version of Theorem 1 with the constant $3C_{b}$ replaced by $C^{\prime}_{b}$.

The algorithm stated in Theorem 1 achieves a lower discrepancy bound than that in Levin’s work [8]. Given a countable set $L$ of positive real numbers greater than $1$, Levin constructs a real number $x$ such that for every $\theta$ in $L$ there is a constant $C^{\prime\prime}_{\theta}$ such that

[TABLE]

The recent analysis in [10] reports no constructions with smaller discrepancy.

For $L=\{2,3,\ldots\}$, Levin’s construction produces a computable sequence of real numbers that converge to an absolutely normal number [1]. To compute the $n$-th term it requires double-exponential in $n$ many operations including trigonometric operations. In contrast, the algorithm presented in Theorem 1 is based just on discrete mathematics and yields the expansion of the computed number by outputting one digit after the other. Unfortunately, to compute the first $n$ digits it performs triple-exponential in $n$ many operations. Thus, the question raised in [4] remains open :

Is there an absolutely normal number computable in polynomial time having a nearly optimal discrepancy of normality ?

Finally we comment that it is possible to prove a version of Theorem 1 replacing the set of integer bases by any countable set of computable real numbers greater than $1$. The proof would remain essentially the same except that one needs a suitable version of Lemma 3.

1 Primary definitions and results

We use some tools from [6] and [9]. For non-negative integers $M$ and $N$, for a sequence of real numbers $(x_{j})_{j\geq 1}$ and for real numbers $\alpha_{1},\alpha_{2}$ such that $0\leq\alpha_{1}<\alpha_{2}\leq 1$, we define

[TABLE]

We write $\mu$ to denote Lebesgue measure.

Lemma 2** **([3, Lemma 8],

adapted from Hardy and Wright [7, Theorem 148]).

Let $b$ be an integer greater than or equal to $2$. Let $m$ and $N$ be positive integers and let $\varepsilon$ be a real such that $6/\lfloor{N/m\rfloor}\leq\varepsilon\leq 1/b^{m}$. Then, for any non-negative integer $M$ and for any integer $a$ such that $0\leq a<b^{m}$,

[TABLE]

is less than $2b^{2m-2}m\ e^{-\varepsilon^{2}Nb^{m}/(6m)}.$

The next lemma is similar to Lemma 2 but it considers dyadic intervals instead of $b$-adic intervals.

Lemma 3**.**

Let $b$ be an integer greater than or equal to $2$, let $k$ and $N$ be positive integers and let $\varepsilon$ be a real. Then, for any pair of integers $M$ and $a$ such that $M\geq 0$ and $0\leq a<2^{k}$,

[TABLE]

is less than $9\cdot 2^{2(k+2)}(k+2)e^{-\varepsilon^{2}Nb^{k+2}/(6(k+2))}$.

Remark 4**.**

In [9], Philipp proves a proposition more general than Lemma 3. His result yields the same order of magnitude but does not make explicit the underlying constant while Lemma 3 does.

Clearly, for arbitrary reals $\alpha_{1},\alpha_{2}$ such that $0\leq\alpha_{1}<\alpha_{2}\leq 1$, for any sequence $(x_{j})_{j\geq 1}$ and for any non-negative integers $M$, $N$ and $k$,

[TABLE]

Lemma 5** ([9, Lemma 4], adapted from [6, Lemma 3.10]).**

Let $b$ be an integer greater than or equal to $2$, let $N$ be a positive integer and let $n$ be such that $2^{n}\leq N<2^{n+1}$. Then, there are integers $m_{1},\ldots,m_{n}$ with $0\leq m_{\ell}\leq 2^{n-\ell}-1$ for $\ell=1,\ldots,n$, such that for any positive integer $h$ and any $a$, with $0\leq a<2^{h}$,

[TABLE]

Let $\eta$ and $\delta$ be positive reals. For each integer $b$ greater than or equal to $2$ and for each positive integer $N$ let

[TABLE]

For integers $b,n,a,h,\ell$ and $m$ such that

[TABLE]

define the following sets

[TABLE]

Lemma 6**.**

Let $\eta$ and $\delta$ be positive real numbers. For each $n>e^{6/(\delta\log 2)}$ and for every $b\geq 2$,

[TABLE]

and there is $n_{0}=n_{0}(\eta,\delta)$ such that

[TABLE]

and such that for every real $x$ outside $\bigcup_{n\geq n_{0}}(G_{b,n}\cup H_{b,n})$,

[TABLE]

where $C_{b}$ is Philipp’s constant, $C_{b}=166+664/(\sqrt{b}-1)$.

Proof.

To bound $\mu G_{b,n}$ we apply twice Lemma 3, first with $N=2^{n}$, $k=(h+1)$ and $\varepsilon=2^{-T(2^{n})/8}\phi(2^{n})2^{-n}$, and then with $N=2^{n}$, $k=T(2^{n})$ and $\varepsilon=2^{-T(2^{n})/8}\phi(2^{n})2^{-n}$. We write $\exp(x)$ to denote $e^{x}$ and we write $T$ instead of $T(2^{n})$. Assuming $n\geq 10$,

[TABLE]

To bound $\mu H_{b,n}$ we apply twice Lemma 3 first letting $N=2^{\ell-1}$, $k=(h+1)$ and $\varepsilon=2^{-h/8}\phi(2^{n})2^{-n}$, and then letting $N=2^{\ell-1}$, $k=T(2^{\ell-1})$ and $\varepsilon=2^{-T(2^{\ell-1})/8}\phi(2^{n})2^{-n}$. Again we write $T$ instead of $T(2^{n})$. Assuming $\log\log(2^{n})\geq 8/\delta^{2}$,

[TABLE]

Thus, there is $n_{0}$ such that for every integer $b$ greater than or equal to $2$,

[TABLE]

It follows from Philipp’s proof of [9, Theorem 1] that for every real $x$ outside $\bigcup_{n\geq n_{0}}(G_{b,n}\cup H_{b,n})$,

[TABLE]

where $C_{b}=166+664/(\sqrt{b}-1)$.∎

2 Proof of Theorem 1

We give an algorithm to compute a real outside the set $\bigcup_{b\geq 2}\bigcup_{n\geq n_{0}}\left(G_{b,n}\cup H_{b,n}\right)$. The technique is similar to that used in the computable reformulation of Sierpinski’s construction given in [2].

The next definition introduces finite approximations to this set. Recall that by Lemma 6, for every integer $b\geq 2$, provided $\delta\geq 1/2$ and $n_{0}=n_{0}(\eta,\delta)\geq e^{6/(\delta^{2}\log 2)}$,

[TABLE]

Definition 7**.**

Fix $\delta=1/2$ and fix $\eta\leq 1/8$. For each integer $b\geq 2$, let $z_{b}$ be the least integer greater than $e^{6/(\delta\log 2)}=e^{12/\log 2}$ such that

[TABLE]

We define

[TABLE]

Observe that $\mu(\Delta)<s<\eta$.

For each $n$, let

[TABLE]

The next propositions follow immediately from these definitions.

Proposition 8**.**

For every $n$, $\mu\left(\Delta-\Delta_{n}\right)\leq r_{n}.$

Proposition 9**.**

For every $n$ and $q$ such that $n\leq q$, $\mu\left(\Delta_{q}-\Delta_{n}\right)\leq r_{n}-r_{q}.$

Proposition 10**.**

For any interval $I$ and any $n$, $\mu\left(\Delta\cap I\right)\leq\mu\left(\Delta_{n}\cap I\right)+r_{n}.$

The proof of Theorem 1 follows from the next lemma.

Lemma 11**.**

There is a computable sequence of nested dyadic intervals $I_{0},I_{1},I_{2},\ldots$ such that for each $n$, $\mu I_{n}=2^{-n}$ and $\mu(\Delta\cap I_{n})<2^{-n}$.

Proof.

Proposition 10 establishes, for any interval $I$ and any $m$,

[TABLE]

Then, to prove the lemma it suffices to give a computable sequence of nested dyadic intervals $I_{0},I_{1},I_{2},\ldots$ such that for each $n$, $\mu I_{n}=2^{-n}$ and $\mu\left(\Delta_{p_{n}}\cap I_{n}\right)+r_{p_{n}}<{2^{-n}}$. We establish

[TABLE]

This value of $p_{n}$ is large enough so that the error $r_{p_{n}}$ is sufficiently small to guarantee that even if all the intervals in $\Delta-\Delta_{p_{n}}$ fall in the half of $I_{n}$ that will be chosen as $I_{n+1}$, $I_{n+1}$ will not be completely covered by $\Delta$. We define the $I_{0},I_{1},\ldots$ inductively.

Base case, $n=0$. Let $I_{0}=\left[0,1\right)$. We need to check that $\mu\left(\Delta_{p_{0}}\cap I_{0}\right)+r_{p_{0}}<2^{0}$. Since $p_{0}=2^{2\cdot 0+2}=4$, $b_{p_{0}}=2$ and $z_{b}=2^{2}/\eta\geq 2^{5}$,

[TABLE]

Since $I_{0}=(0,1)$ and $\Delta_{p_{0}}=\emptyset$, $\Delta_{p_{0}}\cap I_{0}=\emptyset$. Then,

[TABLE]

We conclude $\mu\left(\Delta_{p_{0}}\cap I_{0}\right)+r_{p_{0}}=0+s<\eta<1$.

Inductive case, $n>0$. Assume that for each $m=0,1,\ldots,n-1$,

[TABLE]

where $p_{m}=2^{2m+2}$. Note that for $m=0$, $\sum_{j=1}^{m}$ is the empty sum. We split the interval $I_{n-1}$ in two halves of measure ${2^{-n}}$, given with binary representations of their endpoints as

[TABLE]

Since $I^{0}_{n}\cup I^{1}_{n}$ is equal to interval $I_{n-1}$, we have

[TABLE]

Since $p_{n}\geq p_{n-1}$, we obtain

[TABLE]

Adding $r_{p_{n}}+r_{p_{n}}$ to both sides of this inequality we obtain

[TABLE]

Then, by the inductive condition for $m=n-1$,

[TABLE]

Hence, it is impossible that the terms

[TABLE]

be both greater than or equal to

[TABLE]

Let $d\in\{0,1\}$ be smallest such that

[TABLE]

and define

[TABLE]

To verify that $I_{n}$ satisfies the inductive condition it suffices to verify that

[TABLE]

Developing the definition of $r_{p_{j}}$ we obtain

[TABLE]

Then, using that $\eta<1/8$ we obtain the desired result,

[TABLE]

∎

Let’s see that the number $x=0.d_{1}d_{2}d_{3}$ obtained by the next Algorithm 12 is external to

[TABLE]

Suppose not. Then, there must be an open interval $J$ in $\Delta$ such that $x\in J$. Consider the intervals $I_{1}^{d_{1}},I_{2}^{d_{2}},I_{3}^{d_{3}},\ldots$ By our construction, $x$ belongs each of them. Let $j$ be the smallest index such that $I_{j}^{d_{j}}\subset J$, which exists because the measure of $I_{n}^{d_{n}}$ goes to [math] as $n$ increases. Then $I_{j}^{d_{j}}$ is fully covered by $\Delta$. This contradicts that in our construction at each step $n$ we choose an interval $I_{n}^{b_{n}}$ not fully covered by $\Delta$, because as ensured by the proof of Lemma 11,

[TABLE]

We conclude that $x$ belongs to no interval of $\Delta$. Recall that we fixed $\delta=1/2$; thus, by Lemma 6, for for each integer $b$ greater than or equal to $2$,

[TABLE]

where $C_{b}$ is Philipp’s constant.

Finally, we count the number of mathematical operations that the algorithm performs at step $n$ to compute the digit $d_{n}$ in the binary expansion of $x$. To determine $d_{n}$, the algorithm tests for $d_{n}\in\{0,1\}$ whether

[TABLE]

The naive way to obtain this is by constructing the set $\Delta_{p_{n}}=\bigcup_{b=2}^{b_{p_{n}}}\bigcup_{k=z_{b}}^{p_{n}}G_{b,k}\cup H_{b,k}$, for $b=2,3,\ldots,b_{p_{n}}$. The more demanding is $G_{b_{p_{n}},p_{n}}$ which requires the examination of all the strings of digits in $\{0,\ldots,b_{p_{n}}-1\}$ of length $2^{p_{n}}$. Since $b_{p_{n}}=2n+2$ and $p_{n}=2^{2n+2}$, the number of strings to be examined is

[TABLE]

Thus, with this naive way, the algorithm at step $n$ performs in the order of

[TABLE]

many mathematical operations.

An incremental construction of the sets $G_{b,n}$ and $H_{b,n}$ can lower the number of needed mathematical operations, but would not help to lower the triple-exponential order of computational complexity.

Acknowledgements. We thank Robert Tichy for early discussions on this work. Becher is supported by the University of Buenos Aires and CONICET. Becher is a member of the Laboratoire International Associé INFINIS, CONICET/Universidad de Buenos Aires-CNRS/Université Paris Diderot. Scheerer is supported by the Austrian Science Fund (FWF): I 1751-N26; W1230, Doctoral Program “Discrete Mathematics”; and SFB F 5510-N26. Slaman is partially supported by the National Science Foundation grant DMS-1600441. The present paper was completed while Becher visited the Erwin Schrödinger International Institute for Mathematics and Physics, Austria.