数理解析研究所講究録
第 1874 巻 2014 年 71-80

71

Arithmetical properties of real numbers with low
density of nonzero digits
Hajime Kaneko
JSPS, College of Science and Technology, Nihon University*

1

IntroductionLet

be an integer greater than 1. Borel [2] proved that almost all positive
real numbers are normal in base-b. However, it is generally difficult to show thenormality of a given positive real number . In particular, the base-b expansions of algebraic irrational numbers are mysterious. In this paper, we study
the normality of algebraic irrational numbers. Borel [3] conjectured that all
algebraic irrational numbers are normal in each integral base-b. The conjectureis still an open problem. There is no known example of base-b and positive
irrational such that the normality of in base-b was proven. There is also no
known counterexample of Borel’s conjecture. In particular, it is still not known
whether the digit 1 appears infinitely many times in the decimal expansion of
$b$

$\xi$

$\xi$

$\xi$

$\sqrt{2}.$If Borel’s conjecture is true, then all algebraic irrational numbers are simply
normal in any integral base-b, namely, any letter from the alphabet $\{0,1,$$b-$appears with average frequency tending to $1/b$ . Hence, it is widely believedthat if a positive irrational number has a low density of nonzero digits inbase-b expansion, then is a transcendental number.
In Section 2 we investigate the digits of the base-b expansions of algebraic irrational numbers. In particular, giving lower bounds for the numbers of nonzero
digits of algebraic irrational numbers, we introduce criteria for transcendenceof real numbers. In Section 3 we review -expansions of real numbers, which
gives generalizations of base-b expansions of real numbers. In Section 4 we givemain results on the digits of -expansions of algebraic numbers. Note that if
is a general real number, then the -expansions of rational numbers are also
mysterious. Using the main results, we obtain criteria for transcendence whose-expansion has a low density of nonzero digits. In the last of Section 4 we also
introduce algebraic independence of real numbers with low density of nonzero
digits. In Section 5 we give a sketch of the proof of the main results. Let bea real number. We write the integral and fractional parts of byandrespectively. Moreover, We use the Landau symbols
and the Vinogradovsymbolswith their regular meanings.
$\ldots,$

$1\}$

$\xi$

$\xi$

$\beta$$\beta$$\beta$$\beta$$\beta$

$x$

$x$

$O$
$0,$

$\gg,$ $\ll$*This work is supported by the JSPS fellowship.

$\lfloor x\rfloor$$\{x\},$

72

Base-b expansions of algebraic irrational numbers

2

Let be an integer greater than 1 andbase-b expansion of by
$b$

$\xi$

a positive real number. We write the$\xi$$\xi=\lfloor\xi\rfloor+\sum_{n=1}^{\infty}t_{n}(b;\xi)b^{-n},$

and $t_{n}(b;\xi)\leq b-2$ for infinitely
where $t_{n}(b;\xi)\in\{0,1\ldots, b-1\}$ for any. In this section, we study the
many $n’ s$ . For simphcity, put$\nub(\xi;R)$of digit changes
thenumberanddigitsnumber of nonzero
defined by$n\in \mathbb{Z}^{+}$$t_{0}(b;\xi)$$:=\lfloor\xi\rfloor$$\gamma_{b}(\xi;R)$$\nu b(\xi;R)$$:=$$\gamma_{b}(\xi;R)$$:=$

Card $\{n\in \mathbb{Z}^{+}|n\leq R, t_{n}(b;\xi)\neq 0\},$Card $\{n\in \mathbb{Z}^{+}|n\leq R, t_{n}(b;\xi)\neq t_{n+1}(b;\xi)\},$

respectively, where $R\geq 1$ is a real number and Card denotes the cardinality.
was introduced by Bugeaud [5]. Observe that
The function$\gamma_{b}(\xi;R)$$\nu b(\xi;R)\geq\frac{1}{2}\gamma_{b}(\xi;R)+O(1)$

.

(2.1)

Various mathematicians have studied the digits of algebraic irrational numbers,
using Diophantine approximation methods. We recall Ridout’s theorem [14].
the -adic absolute value, normalized
For any prime number , we denote bydisjoint
finite sets of prime numbers and. Let and besuch thata real algebraic number. Then, for any positive real number , there are only
finitely many rational numbers $p/q$ with $q\geq 1$ such that$l$$|l|\iota=l^{-1}$$|\cdot|\iota$$l$$S_{2}$$S_{1}$

$e$

$\xi$

.

(2.2)

$| \xi-\frac{p}{q}|\cdot\prod_{l\in S_{1}}|p|\iota\cdot\prod_{1\in S_{2}}|q|\iota<\frac{1}{q^{2+\epsilon}}$Let be an algebraic irrational number. Bugeaud [5] pointed out that the
Ridout’s theorem implies
$\xi$

.

(2.3)

$\lim_{Rarrow\infty}\frac{\nu b(\xi;R)}{\log R}=\infty, Rarrow\infty hm\frac{\gamma_{b}(\xi;R)}{\log R}=\infty$

Here, we only check the second inequality of (2.3) in order to introduce a typicalexample of Diophantine approximation methods. Set$S_{1}$$:=\emptyset,$$S_{2}$$:=${ $l|l$ is a prime dividing },$b$$\{n\in \mathbb{N}|t_{n}(b;\xi)\neq 0\}=:\{w_{0}<w_{1}<\cdots\}.$is an infinite sequence. For simplicity,
Since is irrational,$\underline{w}_{m}(m=0,1,$
we put $t_{w_{m}}(b;\xi)=:t_{m}$ . Then we have $t_{m}\in\{1,2, \ldots , b-1\}$ for any $m\geq 1$ and$\xi$$\ldots\underline{)}$$\xi=\sum_{m=0}^{\infty}\overline{t_{m}}b^{-w_{m}}.$73
We apply the Ridout’s theorem with
$p=b^{w_{M}} \sum_{m=0}^{M}\overline{t_{m}}b^{-w_{m}}, q=b^{w_{M}}$for sufficiently large $M$ . Observe that$\prod_{l\in S_{2}}|q|_{l}=q^{-1}=b^{-w_{M}}.$Let be an arbitrary positive real number. Since (2.2) has only finitely manysolutions of rational numbers, we get, for any sufficiently large $M,$$\epsilon$$b^{-(1+\epsilon)w_{M}}$$=$$q^{-(1+\epsilon)} \leq|\xi-\frac{p}{q}|$$= \sum_{m=M+1}^{\infty}\overline{t_{m}}b^{-w_{m}}\leq\sum_{h=0}^{\infty}(b-1)b^{-w_{M+1}-h}=b^{1-w_{M+1}}.$Thus,$w_{M+1}\leq 1+(1+\epsilon)w_{M}\leq(1+2\epsilon)w_{M}$for any sufficiently large $M$ . Since$\epsilon$

is arbitrary, we deduce that$\lim_{Marrow\infty}\frac{w_{M+1}}{w_{M}}=1,$

which imphes the second equality of (2.3).
Applying a quantitative Ridout’s theorem [11], Bugeaud [5] improved (2.3)

as follows:$\gamma_{b}(\xi;R)\geq 3(\log R)^{1+1/(\omega(b)+4)}$$($

log log $R)^{-1/4}$

for any sufficiently large , whereis the number of the distinct prime
factors of . In particular, (2.1) implies that$R$$\omega(b)$$b$$\nu b(\xi;R)\geq(\log R)^{1+1/(\omega(b)+4)}(\log\log R)^{-1/4}$

for any sufficiently large . Improving the quantitative parametric subspace
theorem by Evertse and Schlickewei [8], Bugeaud and Evertse [7] proved thefollowing: there exists an effectively computable positive constant, depending only on , such that if is an algebraic irrational number of degree $D,$
then
$R$

$C_{1}(\xi)$

$\xi$

$\xi$

$\gamma_{b}(\xi;R)\geq C_{1}(\xi)(\log R)^{3/2}(\log\log R)^{-1/2}$(2.4)

for any sufficiently large . Bailey, Borwein, Crandall, and Pomerance [1] gave
a new method to estimate the lower bounds for $vb(\xi;N)$ . We mention themethod in Section 5. Let again be an algebraic irrational number of degree $D.$
They showed that if $b=2$ , then there exists an effectively computable positive, depending only on , satisfying
constant
$R$

$\xi$

$C_{2}(\xi)$$\xi$$\nu_{2}(\xi;R)\geq C_{2}(\xi)R^{1/D}$

(2.5)

74
for any sufficiently large . Modifying their method, we can generalize (2.5) for
general integral base as follows: there exists an effectively computable positive, depending only on and , such that
constant
$R$

$b$

$b$

$C_{2}’(b;\xi)$

$\xi$

(2.6)

$\nu_{b}(\xi;R)\geq C_{2}’(b;\xi)R^{1/D}$

for any sufficiently large $R$ . For instance, see Theorem 8.5 in [4]. Inspiredby the method in [1], the author [9, 10] improved (2.4) for certain classes ofalgebraic irrational of degree $D$ . Namely, if satisfies certain assumptions
on its minimal polynomial, then there exists an effectively computable positive
$\xi$

$\xi$

constant$C_{3}(b;\xi)$such that$\gamma_{b}(\xi;R)\geq C_{3}(b;\xi)R^{1/D}$

for any sufficiently large $R.$
In the rest of this section we apply (2.6) to transcendence of real numberswith low density of nonzero digits. Let $w=(w_{m})_{m=0}^{\infty}$ be a sequence of nonnegative integers such that $w_{m+1}>w_{m}$ for any sufficiently large $m$ . Put$f(w;z):= \sum_{m=0}^{\infty}z^{w_{m}}$

.

(2.7)

If Borel’s conjecture for the normality of algebraic irrational numbers is true,
then we obtain the following criteria: If $w$ satisfies$\lim_{marrow\infty}\frac{w_{m}}{m}=\infty,$

then $f(w;b^{-1})= \sum_{m=0}^{\infty}b^{-w_{m}}$ is transcendental. However, it is unknown whether
the criteria above hold. On the other band, using (2.6), we deduce partial results. Namely, assume that $w$ satisfies(2.8)$\lim_{marrow\infty}\frac{w_{m}}{m^{A}}=\infty$

for any positive real number
put

$A$

. Then $f(w;b^{-1})$ is transcendental. For example,$\varphi_{y}(m) := \lfloor m^{(\log m)^{y}}\rfloor=\lfloor\exp((\log m)^{1+y})\rfloor$

,

(2.9)
(2.10)$\mu_{y}(z) := \sum_{rn=1}^{\infty}z^{\varphi_{y}(m)}$for a positive real number . Then, since $\varphi_{y}(m)(m=1,2, \ldots)$ satisfies (2.8),
we deduce thatis transcendental for any integer $b\geq 2.$$y$$\mu_{y}(b^{-1})$

3$\beta$-expansions of real numbers

be a real number greater than 1. The notion of the -expansions ofLetreal numbers was introduced by R\’enyi [13]. Let: $[0,1)arrow[0,1)$ be thefor $x\in[O, 1)$ . Let $\eta\in[0,1)$ . Then thetransformation defined by-expansion of is denoted by$\beta$$\beta$$T_{\beta}$$T_{\beta}(x)$$\beta$$\beta-$$:=\{\beta x\}$$\eta$$\eta=\sum_{n=1}^{\infty}t_{n}(\beta;\xi)\beta^{-n}$

,

(3.1)

75
where
for any. If $\beta=b$ is an integer
greater than 1, then (3.1) gives the base-b expansion of
Recall that the base-b expansion of each rational number is ultimately periodic. Schmidt [15] studied the periodicity of the -expansions of rational
numbers. We call $\beta>1$ Pisot number if is an algebraic integer such thatthe conjugates except itself have moduli less than 1, Moreover, we say that$\beta>1$ is a Salem number if
is an algebraic integer such that the conjugates
except itself have moduli at most 1 and that at least one conjugate has modulus
1. Schmidt [15] showed that if the -expansion of each rational number is ultimately periodic, then is a Pisot or Salem number. Moreover, he showed that if
is a Pisot number, then the -expansion of any rational number is ultimately
periodic. However, if is a Salem number, then the -expansions of rational
numbers are mysterious. Schmidt conjectured that if is a Salem number, thenthe -expansion of every rational number is ultimately periodic, which is still
an open problem. In the next section, we investigate the -expansions of rational numbers and more general algebraic numbers in connection with Schmidt’s
conjecture and Borel’s conjecture.
In the rest of this section, we recall the expansion of 1. The expansion of 1
is defined by$n\in \mathbb{Z}+$$t_{n}(\beta;\eta)=\lfloor\beta T_{\beta}^{n-1}(\eta)\rfloor\in \mathbb{Z}\cap[0, \beta)$$\eta.$$\beta$$\beta$$\beta$$\beta$$\beta$$\beta$$\beta$$\beta$$\beta$$\beta$$\beta$$\beta$$1= \sum_{n=1}^{\infty}t_{n}(\beta;1-)\beta^{-n},$where$t_{n}( \beta;1-) :=\lim_{xarrow 1-}t_{n}(\beta;x)\in \mathbb{Z}\cap[O, \beta)$.

The expansion of 1 has a crucial role for studying the -expansions of real
numbers. Moreover, the periodicity of the expansion of 1 is important for investigating the -shifts. Parry [12] showed that if is a Pisot number, then the
expansion of 1 is ultimately periodic. We call that a Parry number if the expansion of 1 is ultimately periodic. However, it is unknown whether there exists
a non-Parry Salem number. We investigate the periodicity of the expansion of
1 in Section 4.
$\beta$$\beta$$\beta$$\beta$4

Main results

We now introduce main results which gives arithmetical properties of the valuesof power series at certain algebraic points. Let
be a boundedsequence of nonnegative integers. Put$s=(s_{n})_{n=0}^{\infty}$$g(s;z):= \sum_{n=0}^{\infty}s_{n}z^{n}$and, for a nonnegative real number $R,$$\lambda(s;R) :=Card\{n\in \mathbb{N}|n\leq R, s_{n}\neq 0\}.$Moreover, let $K$ and be algebraic number fields with
by $[L : K]$ the degree of field extension.
$L$$K\subset$

L. Then we denote

76
THEOREM 4.1. Let be a Pisot or Salem number and an algebraic number
withLet
be a sequence of integers with
$0\leq s_{n}\leq B$, whereis a positive integer independent offor any. Suppose that $s_{n}\neq 0$ for infinitely many $ns$ . Then there exist effectively, depending only on
computable positive constantsand, and , such that$\beta$$[\mathbb{Q}(\beta, \xi) :$\xi$$s=(s_{n})_{n=0}^{\infty}$\mathbb{Q}(\beta)]=D.$$B$$n\in \mathbb{N}$$n$$C_{4}(\beta, \xi, B)$$\beta,$$C_{5}(\beta, \xi, B)$

$B$

$\xi$

$\lambda(s;R)\geq C_{4}(\beta, \xi, B)R^{1/(-1+2D)}(\log R)^{-1/(-1+2D)}$

for any real number$R$with $R\geq C_{5}(\beta,\xi, B)$ .

We apply Theorem 4.1 to the transcendence of $f(w;\beta^{-1})$ , where $f(w;z)$ isdefined by (2.7) and is a Pisot or Salem number.
$\beta$COROLLARY 4.2. Let
be a sequence of nonnegative integers
such that $v_{m+1}>v_{m}$ for any sufficiently large $m$ . Suppose for any positive real
number that$w=(w_{m})_{m=0}^{\infty}$$A$$\lim_{marrow\infty}\frac{w_{m}}{m^{A}}=\infty.$

Then $f(w;\beta^{-1})$ is transcendental for any Pisot or Salem number$\beta.$Let be a positive real number. Recall thatandare defined by(2.9) and (2.10), respectively. Corollary 4.2 implies thatis transcendental for any Pisot or Salem number
We apply Theorem 4.1 to the -expansions of algebraic numbers $\eta\in[0,1)$ .
In the rest of this section, the imphed constants in the symbol
are effectively
computable ones depending only on and . We generalize the notation in
Section 2. Namely, put$\mu_{y}(z)$$\varphi_{y}(m)$$y$$\mu_{y}(\beta^{-i})$$\beta.$$\beta$$\gg$$\beta$$\eta$$\nu_{\beta}(\eta;R) := Card\{n\in \mathbb{Z}^{+}|n\leq R, t_{n}(\beta;\eta)\neq0\}$$\gamma_{\beta}(\eta;R)$

Card $\{n\in \mathbb{Z}^{+}|n\leq R, t_{n}(\beta;\eta)\neq t_{n+1}(\beta;\eta)\}.$$:=$

Then we have$\nu_{\beta}(\eta;R)\geq\frac{1}{2}\gamma_{\beta}(\eta;R)+O(1)$

.

(4.1)

be a Pisot or Salem number and $\eta\in[0,1)$ an algebraic number with
. Bugeaud [6] showed that ifforinfinitely many $n’ s$ , thenLet$\beta$$[\mathbb{Q}(\beta, \eta) :$t_{n}(\beta;\eta)\neq t_{n+1}(\beta;\eta)$\mathbb{Q}(\beta)]=D$$\gamma_{\beta}(\eta;R)\gg(\log R)^{3/2}($

for any sufficiently large$R$. Moreover, (4.1) imphes that$\nu_{\beta}(\eta;R)\gg(\log R)^{3/2}($

for any sufficiently large$R$

log log $R)^{-1/2}$

log log $R)^{-1/2}$(4.2). On the other hand, using Theorem 4.1, we obtain$\nu_{\beta}(\eta;R)\gg R^{1/(-1+2D)}(\log R)^{-1/(-1+2D)}$for every sufficiently large $R$ , which gives improvement of (4.2).

(4.3)

77
We now consider the Schmidt conjecture on the periodicity of rational numbers. Suppose that is a Salem number and that $\eta\in[0,1)$ is a rational number.
Iffor infinitely many ’s and if the sequence $t_{n}(\beta;\eta)(n=1,2, \ldots)$
is ultimately periodic, then we have $\nu_{\beta}(\eta;R)\gg R$ . Now, (4.3) means that$\beta$$t_{n}(\beta;\eta)\neq 0$$n$$\nu_{\beta}(\eta;R)\gg R(\log R)^{-1}$for each sufficiently large , which gives partial results on the Schmidt’s conjecture.
Theorem 4.1 is applicable to the study of the expansion of 1. It is well-known
for any real number $\beta>1$ that $t_{n}(\beta;1-)\neq 0$ for infinitely many $n’ s$ . Let again
be a Salem number. Put$R$$\beta$$\nu_{\beta}(1-;R) :=Card\{n\in \mathbb{Z}^{+}|t_{n}(\beta;1-)\neq 0\}.$If the expansion of 1 is ultimately periodic, then we have $\nu_{\beta}(1-;R)\gg R.$
Bugeaud [6] showed that$\nu_{\beta}(1-;R)\gg(\log R)^{3/2}(\log\log R)^{-1/2}$

for any sufficiently large $R$ . On the other hand, using Theorem 4.1, we deducethat$\nu_{\beta}(1-;R)\gg R(\log R)^{-1}$for each sufficiently large $R.$

5

Sketch of the proof of main results

In this section we introduce ideas for the proof of the main results. For simplicity,
we give a sketch of the proof of Corollary 4.2. We define
and by$s=(s_{n})_{n=0}^{\infty}$

$\xi$

,

$f(w;z)= \sum_{m=0}^{\infty}z^{w_{m}}=:\sum_{n=0}^{\infty}s_{n}z^{n}, \xi:=f(w;\beta^{-1})$

respectively. Then is bounded and. We use the samefor any
notation as in Section 4. Then we have, for any positive real number$s$$n\in \mathbb{N}$$s_{n}\in \mathbb{N}$$\epsilon,$$\lambda(s;R)=o(R^{\epsilon})$as

tends to infinity. We show that $P(\xi)\neq 0$ , where $P(X)=A_{D}X^{D}+$is any non-constant polynomial with $A_{D}\neq 0$ . In
what follows,
and the the imphed constants in the symbols
arepositive constants depending only on $w$ and $P(X)$ . For instance, $0\leq s_{n}\leq C_{6}$for any. Put. Without loss of generality, we may
assume that
. We now calculate
for any with $1\leq k\leq D$ . Then we$R$$A_{D-1}X^{D-1}+\cdots+A_{0}\in \mathbb{Z}[X]$$C_{6},$ $C_{7}\ldots$$n\in \mathbb{N}$$0\in\Gamma$$\Gamma$$\ll,$ $\gg$$:=\{n\in \mathbb{N}|\mathcal{S}_{n}\neq 0\}$$\xi^{k}$

$k$

78
get
$\xi^{k} = (\sum_{m\in\Gamma}s_{m}\beta^{-m})^{k}$$= \sum_{m_{1},\ldots,mk\in\Gamma}s_{m_{1}}\cdots s_{m_{k}}\beta^{-m_{1}-\cdots-m_{k}}$$= m=0m_{1,.\cdot.\cdot.\cdot k} \sum^{\infty}\beta^{-m}\sum_{m_{k}m_{1+\dotplus^{m\in\Gamma}}=n}s_{m_{1}}\cdots s_{m}k$$=$:$\sum_{m=0}^{\infty}\beta^{-m}\rho(k;m)$

,

where
$\rho(k;m)=\sum_{1m,.\cdot.\cdot.\cdot mk\in\Gamma m1+\dotplus_{m_{k}=m}}s_{m_{1}}\cdots s_{m_{k}}.$. Then $\rho(k;m)$ is a nonnegative integer because
Let
. Set
integer for any$m\in \mathbb{N}$$s_{n}$

is a nonnegative$n\in \mathbb{N}$$k\Gamma:=\{+\cdots+m|m_{1}, \ldots, m_{k}\in\Gamma\}.$Since$0\in\Gamma$, we have$\Gamma\subset 2\Gamma\subset\cdots\subset(D-1)\Gamma\subset D\Gamma.$

Observe that $\rho(k;m)$ is positive if and only if $m\in kS$ . Now we introduce BBP. Usingtails. Let$R\in \mathbb{N}$$P( \xi) = A_{0}+\sum_{k=1}^{D}A_{k}\xi^{k}$$= A_{0}+ \sum_{k=1}^{D}A_{k}\sum_{m=0}^{\infty}\beta^{-m}\rho(k;m)$,

we obtain$\beta^{R}P(\xi) = A_{0}\beta^{R}+\sum_{k=1}^{D}A_{k}\sum_{m=0}^{\infty}\beta^{-(m-R)}\rho(k;m)$$= A_{0} \beta^{R}+\sum_{k=1}^{D}A_{k}\sum_{m=-R}^{\infty}\beta^{-m}\rho(k;m+R)$

Put$Y_{R} ;= \sum_{k=1}^{D}A_{k}\sum_{m=1}^{\infty}\beta^{-m}\rho(k;m+R)$,.

$Z_{R} := A_{0} \beta^{R}+\sum_{k=1}^{D}A_{k}\sum_{m=-R}^{0}\beta^{-m}\rho(k;m+R)$.

79
Then we have$\beta^{R}P(\xi)=Y_{R}+Z_{R}$

.

(5.1)

Note thatis an algebraic integer. In the paper [1],is called BBP tails
in the case of $\beta=2$ . For the proof of $P(\xi)\neq 0$ , we consider BBP tails in thecase where is a Pisot or Salem number. In what follows, we estimate
and, respectively. If $\beta=2$ , then one of the key ideas for the proof of (2.5) is thefollowing:$Z_{R}$$Y_{R}$$\beta$$Y_{R}$$Z_{R}$

If $Z_{R}\neq 0$ , then $|Z_{R}|\geq 1$ .

In the case where$\beta$(5.2)

is a general Pisot or Salem number, we get the following:

If $Z_{R}\neq 0$ , then $|Z_{R}|\geq C_{7}R^{-C_{8}}$ .

(5.3)

Put
$y_{N}:=$

Card $\{R\in \mathbb{N}|R\leq N, Y_{R}\neq 0\}$for positive integer $N$ . Bailey, Borwein, Crandall, and Pomerance [1] estimated, using the relation (5.2). However, (5.3) does not seem enough for theestimation of . Hence, we calculate. Division of the interval of $[0, N)$ into
subintervals is also one of the key ideas for the proof of (2.5).
In the rest of this section, we give an outhne. We write the length of aninterval
by $|[x, y)|=y-x$ . First, using a combinatorial method, weconstruct an subinterval $J\subset[O, N)$ satisfying$y_{N}$$y_{N}$$Y_{R}$$[x, y)\in \mathbb{R}$$|J| \gg\frac{N}{\lambda(s;N)^{D-1}}$

and $Y_{R}>0$ for any $R\in J$ . Next, we divide
we constructfulfilling$J$

into subintervals. Consequently,$I\subset J$$|I| \gg\frac{N}{\lambda(s;N)^{-1+2D}}$

and$0<Y_{R}< \frac{1}{2}C_{7}R^{-C_{8}}$(5.4)

for any with $R\in I$ . Hence, if $R\in I$ , then combining (5.1), (5.3), and (5.4),
we deduce that $P(\xi)\neq 0.$$R$

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