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MATHEMATICS
M. F. KULIKOVA
CONSTRUCTION OF A NUMBER \(\alpha\) FOR WHICH THE FRACTIONAL PARTS \(\{\alpha g^x\}\) ARE RAPIDLY UNIFORMLY DISTRIBUTED
(Presented by Academician I. M. Vinogradov, November 22, 1961)
Let \(g \geq 2\) be an integer, \(\alpha\) a real number, \(\mathfrak M\) an interval lying on the half-segment \([0,1)\), \(\operatorname{mes}\mathfrak M\) its length, and \(N_P(\mathfrak M)\) the number of numbers \(\{\alpha g^x\}\), \(x=1,2,\ldots,P\), that fall in the interval \(\mathfrak M\).
N. M. Korobov \((^1)\) constructed a number \(\alpha\) such that, for any interval \(\mathfrak M\) located in the interval \([0,1)\), the estimate
\[ N_P(\mathfrak M)=P\,\operatorname{mes}\mathfrak M+O(\sqrt P), \]
holds, where \(\operatorname{mes}\mathfrak M\) is the length of the interval \(\mathfrak M\).
A. G. Postnikov \((^2)\), using the method of trigonometric sums, constructed a number \(\alpha\) such that, for any interval \(\mathfrak M\) located in the interval \([0,1)\), the estimate
\[ N_P(\mathfrak M)=P\,\operatorname{mes}\mathfrak M +O\left(\frac{\sqrt P}{\sqrt[8]{\log P}}\log\log P\right) \]
holds.
In the present note we construct a number \(\alpha\) such that, for any interval \(\mathfrak M\) located in \([0,1)\), the following estimate will hold:
\[ N_P(\mathfrak M)=P\,\operatorname{mes}\mathfrak M +O\left(\frac{\sqrt P}{\sqrt[4]{\log P}}(\log\log P)^{3/2}\right). \]
We follow N. M. Korobov’s idea of constructing numbers \(\alpha\) with the help of normal periodic systems \((^3)\). Denote by \(\rho_n(g)\) a normal periodic system, and by \(\rho'_n(g)\) the system obtained from \(\rho_n(g)\) by discarding the last \(n-1\) digits. Let
\[ \varphi(j)=\left[\frac{g^j}{\sqrt j}\right], \qquad j=1,2,\ldots . \]
Consider the number \(\alpha\) defined in the following way:
\[ \alpha=0,\, \underbrace{\rho'_1(g)\ldots \rho'_1(g)}_{\varphi(1)\ \text{times}}\, \cdots\, \underbrace{\rho'_\mu(g)\ldots \rho'_\mu(g)}_{\varphi(\mu)\ \text{times}}\, \cdots \tag{1} \]
Theorem. Let the number \(\alpha\) be defined by the string (1). For any interval \(\mathfrak M\) located in \([0,1)\), the formula
\[ N_P(\mathfrak M)=P\,\operatorname{mes}\mathfrak M +O\left(\frac{\sqrt P}{\sqrt[4]{\log P}}(\log\log P)^{3/2}\right) \]
holds.
Proof. It suffices to prove the theorem for intervals \(\mathfrak M\) of the form \((0,y)\), \(0<y<1\). Denote by \(S_\mu\) the number of \(g\)-ary digits of the number \(\alpha\) up to the beginning of the first system \(\rho'_\mu(g)\). Denote by \(S_{\mu,\lambda}\) the number of \(g\)-ary digits in the number \(\alpha\) up to the beginning of the \(\lambda\)-th system \(\rho'_\mu(g)\) in order (obviously, \(S_\mu=S_{\mu,1}\)):
\[ S_{\mu,\lambda}=\sum_{j=1}^{\mu-1}\varphi(j)g^j+(\lambda-1)g^\mu . \]
Arguing as in (²), we obtain
\[ N_{S_{\mu,\lambda}}(\mathfrak M)=yS_{\mu,\lambda}+O\left(\frac{g^\mu}{\sqrt{\mu}}\right). \]
Let us note that
\[ \mu=\frac12\frac{\log P}{\log g}+\frac14\frac{\ln g\,\log P}{\log g}+O(1). \]
Let
\[ P=S_{\mu,\lambda}+P_1,\qquad 0\leq P_1\leq g^\mu-1, \]
\[ N_P(\mathfrak M)=YP+N_P(\mathfrak M)-N_{S_{\mu,\lambda}}(\mathfrak M)-YP_1+O\left(\frac{g^\mu}{\sqrt{\mu}}\right). \]
Two cases may occur:
1) \(0\leq P_1\leq \dfrac{g^\mu}{\sqrt{\mu}}\). Then
\[ N_P(\mathfrak M)-N_{S_{\mu,\lambda}}(\mathfrak M)-YP_1=O\left(\frac{g^\mu}{\sqrt{\mu}}\right). \]
2) In the case \(g^\mu>P_1\geq \dfrac{g^\mu}{\sqrt{\mu}}\) we shall rely on the following lemma.
Lemma. Let \(\mathfrak M\) be any interval of the form \((0,Y)\), \(0<y<1\). Let \(n,l\) be natural parameters, \(4g^n\leq l\leq \dfrac{P_1-1}{n}\). \(N_{P_1}(\mathfrak M)\) denotes the number of hits of the fractional parts \(\{\omega g^x\}\), \(x=1,2,\ldots,P_1\), in \(\mathfrak M\). For sufficiently large \(P_1\) the estimate
\[ \left|N_P(\mathfrak M)-YP\right| <C\left(g^nlnl\sqrt{l}\,N_{P_1}^{(nl)} \exp\left[-\frac{l}{18g^{2n}}\right]+ln+\frac{P_1}{g^n}\right), \]
holds, where \(C\) is an absolute constant.
The proof of this lemma is carried out by the method proposed by I. I. Piatetski-Shapiro (⁴). We shall not give the proof of this lemma.
Take as the number \(\omega\) the number
\[ \omega=\{g^{S_{\mu,\lambda}}\alpha\}. \]
The \(g\)-adic expansion of the number \(\omega\) has the form
\[ \omega=\underbrace{\rho'_\mu(g)\ldots\rho'_\mu(g)}_{\varphi(\mu)-(\lambda-1)\text{ times}}\quad \underbrace{\rho'_{\mu+1}(g)\ldots\rho'_{\mu+1}(g)\ldots}_{\varphi(\mu+1)\text{ times}} \]
Take
\[ n=\left[\frac{\log\sqrt{\dfrac{\log P_1}{(\log\log P_1)^3}}}{\log g}\right], \qquad l=\left[\frac{\mu}{n}\right]+1. \]
Applying the lemma, we obtain
\[ \left|N_P(\mathfrak M)-N_{S_{\mu,\lambda}}(\mathfrak M)-YP_1\right|\leq \]
\[ \leq C\left(g^\mu\left(\frac{\log P_1}{\log\log P_1}\right)^2 N_{P_1}^{(nl)}e^{-C_0(\log\log P_1)^2} +\frac{P_1(\log\log P_1)^{3/2}}{\sqrt{\log P_1}}\right)\leq \]
\[ \leq Cg^\mu\left(\left(\frac{\log P_1}{\log\log P_1}\right)^2 N_{P_1}^{(nl)}e^{-C_0(\log\log P_1)^2} +\frac{(\log\log P_2)^{3/2}}{\sqrt{\log P_1}}\right). \]
Since \(nl \geqslant \mu\), and the first \(g^\mu\) \(g\)-ary digits of \(\omega\) form the system \(\hat\rho_\mu(g)\), it follows that \(N_{P_1}^{(nl)}=1\). For sufficiently large \(P_1\),
\[ \left(\frac{\log P_1}{\log\log P_1}\right)^2 e^{-C_0(\log\log P_1)^2} < \frac{(\log\log P_1)^{3/2}}{\sqrt{\log P_1}} . \]
Taking into account that \(C_2\mu \leqslant \log P_1 \leqslant C_1\mu\), where \(C_1\) and \(C_2\) are positive constants, we obtain
\[ N_P(\mathfrak M)-N_{S_\mu,\lambda}(\mathfrak M)-Y P_1 = O\left( \frac{g^\mu(\log\mu)^{3/2}}{\sqrt{\mu}} \right). \]
Thus, in both cases
\[ N_P(\mathfrak M)-YP = O\left( \frac{g^\mu(\log\mu)^{3/2}}{\sqrt{\mu}} \right). \]
Since
\[ \mu= \frac{1}{2}\frac{\log P}{\log g} + \frac{1}{4}\frac{\log\log P}{\log g} + O(1), \]
we have
\[ N_P(\mathfrak M) = YP+ O\left( \frac{\sqrt P}{\sqrt[4]{\log P}}(\log\log P)^{3/2} \right). \]
Received
17 XI 1961
REFERENCES
- N. M. Korobov, Izv. Akad. Nauk SSSR, Ser. Mat., 19, 361 (1955).
- A. G. Postnikov, Uspekhi Mat. Nauk, 16, no. 3 (99), 201 (1961).
- N. M. Korobov, Izv. Akad. Nauk SSSR, Ser. Mat., 14, 215 (1950).
- I. I. Pyatetskii-Shapiro, Uch. Zap. Mosk. Gos. Ped. Inst. im. V. I. Lenina, 108, no. 2, 317 (1957).