Abstract
Full Text
Mathematics
V. G. SPRINDZHUK
ON MAHLER’S HYPOTHESIS
(Presented by Academician I. M. Vinogradov on 30 IX 1963)
Let \(\omega\) be a transcendental number. For a given positive integer \(n\), define \(w_n(\omega)\) as the exact upper bound of those numbers \(w\) for which there exist infinitely many integral polynomials \(P\) of degree not exceeding \(n\) satisfying the inequality
\[ |P(\omega)|<h^{-w}\quad (h\to\infty), \tag{1} \]
where \(h\) is the height of the polynomial \(P\). Put
\[ \frac{1}{n}w_n(\omega)= \begin{cases} \theta_n(\omega), & \text{if } \omega \text{ is real},\\ \eta_n(\omega), & \text{if } \omega \text{ is complex}. \end{cases} \]
It is well known \(\left({}^{4},\text{ p. }96\right)\) that
\[ \theta_n(\omega)\geqslant 1,\quad \eta_n(\omega)\geqslant \frac{1}{2}-\frac{1}{2n} \quad (n=1,2,\ldots) \tag{2} \]
for all transcendental numbers \(\omega\). On the other hand, K. Mahler proved \(\left({}^{1}\right)\) that for almost all (in the sense of Lebesgue measure) real and complex numbers
\[ \theta_n(\omega)\leqslant 4,\quad \eta_n(\omega)\geqslant \frac{7}{2} \quad (n=1,2,\ldots), \tag{3} \]
and conjectured that in the last inequalities one may take \(1\) and \(1/2\), respectively, instead of \(4\) and \(7/2\). Thus, according to Mahler’s hypothesis and by virtue of (2),
\[ \sup_{(n)} \theta_n(\omega)=1,\quad \sup_{(n)} \eta_n(\omega)=\frac{1}{2} \quad (n=1,2,\ldots) \]
for almost all numbers \(\omega\).
Koksma proved \(\left({}^{2}\right)\) that Mahler’s constants \(4\) and \(7/2\) can be replaced by \(3\) and \(5/2\), respectively; then LeVeque \(\left({}^{3}\right)\) obtained an analogous result for the constants \(2\) and \(3/2\). Finally, B. Folkman \(\left({}^{7}\right)\) obtained the inequalities
\[ \theta_n(\omega)\leqslant \frac{3}{2},\quad \eta_n\leqslant \frac{3}{4}-\frac{1}{2n} \quad (n=1,2,\ldots) \]
for almost all numbers.
The author of the present paper proved \(\left({}^{11}\right)\) the existence of such numbers \(\theta_n,\eta_n\) that, for almost all \(\omega\),
\[ \theta_n(\omega)=\theta_n,\quad \eta_n(\omega)=\eta_n \quad (n=1,2,\ldots) \]
and obtained \(\left({}^{12}\right)\) the inequalities
\[ \theta_n<\frac{4}{3},\quad \eta_n<\frac{2}{3} \quad (n=1,2,\ldots). \]
In view of the results of Kubilius \(\left({}^{10}\right)\), Folkman \(\left({}^{5,6}\right)\), and Cassels \(\left({}^{8}\right)\), the equalities
\[ \theta_2=\theta_3=1,\quad \eta_2=\frac{1}{4},\quad \eta_3=\frac{1}{3}. \tag{4} \]
hold.
In this paper we briefly set forth the scheme of the proof of Mahler’s hypothesis for complex numbers, i.e., the scheme of the proof of the inequalities
\[ \frac{1}{2}-\frac{1}{2n}\leqslant \eta_n\leqslant \frac{1}{2} \quad (n=1,2,\ldots). \tag{5} \]
Analogously, the inequalities
\[ 1 \ll \theta_n \ll 1+\frac{1}{n!}\qquad (n=1,2,\ldots) \]
are proved. Similar results are valid for fields of \(p\)-adic numbers and formal power series (for an introduction see [9, 13]). The method of reasoning is based on the ideas and results of the author’s papers [11, 12].
Lemma 1. Let \(P\) be a polynomial of degree \(n\), height \(h\), without multiple roots \(\chi_1,\chi_2,\ldots,\chi_n\); let \(\omega\) be a complex number,
\[ |\omega-\chi_1|=\min_{(i)}|\omega-\chi_i|\qquad (i=1,2,\ldots,n). \]
Then, if \(|P(\omega)|<h^{-w}\), then
\[ |\omega-\chi_1|^2<c(n,\operatorname{Im}\omega)\, h^{-5/3-\frac{4w-n}{3}}\,|D(P)|^{-1/6}, \]
where \(D(P)\) is the discriminant of \(P\).
For the proof see [12].
Lemma 2. Let \(\Delta\) be a measurable set in the plane, \(\operatorname{mes}\Delta<\varepsilon\).
Let a system \(\Lambda=\bigcup_{i=1}^{\infty}\lambda_i\) of bounded simply connected domains \(\lambda_i\) be given with the conditions
\[ \operatorname{mes}(\lambda_i\cap\Delta)\ge \frac{1}{2}\operatorname{mes}\lambda_i,\qquad \operatorname{mes}\lambda_i>c d_i^2\qquad (i=1,2,\ldots), \]
where \(d_i\) is the diameter of the set \(\lambda_i\), \(c>0\) is a constant. Then \(\operatorname{mes}\Lambda<c_1\varepsilon\), where \(c_1\) is a constant.
The proof of this lemma is elementary, and we omit it.
We pass to the proof of the right-hand side of inequality (5). It is enough to consider the set \(\mathcal P_n(h)\) of irreducible polynomials \(P=a_0+a_1x+\cdots+a_nx^n\) satisfying
\[ \max(|a_0|,\ldots,|a_{n-1}|)\ll a_n=h \]
and to assume that \(\omega\in\Omega\), where \(\Omega\) is a bounded domain in the upper half-plane [12].
Let \(\sigma(P)\) be the set of all \(\omega\) satisfying (1) with some \(w=w_0=w_{n-1}+\delta\), \(\delta>0\), where \(w_n=n\eta_n\) \((n=2,3,\ldots)\). It is clear that \(\sigma(P)\) is a system of at most \(n\) nonintersecting simply connected domains. Let \(\sigma_1(P)\) be one such domain. It is possible that \(\sigma_1(P)\) contains points \(\omega_0\) belonging to systems \(\sigma(Q)\), \(Q\in\mathcal P_n(h)\), \(Q\ne P\). For such a point \(\omega_0\), \(|R(\omega_0)|<2h^{-w_0}\), where \(R=P-Q\ne0\) is a polynomial of degree at most \(n-1\), height at most \(2h\). We say that the domain \(\sigma_1(P)\) is essential if in it the set of points \(\omega_0\) with the indicated property has measure less than \(\frac12\operatorname{mes}\sigma_1(P)\), and inessential in the opposite case. It can be shown that the domains \(\sigma_1(P)\) have the property \(\operatorname{mes}\sigma_1(P)>c(n)d^2\), where \(d\) is the diameter of the domain \(\sigma_1(P)\). This is derived with the aid of Lemma 1 from the fact that under the conditions of this lemma, for \(w\ge w_{n-1}\),
\[ |\omega-\chi_1|\asymp \begin{cases} |P(\omega)|:|P'(\chi_1)|, & \text{if } |\omega-\chi_1|\le 2|\chi_1-\chi_2|,\\[4pt] \bigl(|P(\omega)|\,|\chi_1-\chi_2|:|P'(\chi_1)|\bigr)^{1/2}, & \text{if } |\omega-\chi_1|>2|\chi_1-\chi_2|, \end{cases} \]
where \(\chi_2\) is the root of \(P\) nearest to \(\omega\) after \(\chi_1\). Applying Lemma 2 to the inessential domains, we find that the measure of the set of points falling into infinitely many inessential domains is zero for any \(\delta>0\).
We note that the number of polynomials \(P\) having an essential domain \(\sigma_1(P)\) with the condition \(\operatorname{mes}\sigma_1(P)\ge\lambda>0\) will be \(\ll \lambda^{-1}\).
The domain \(\sigma_1(P)\) contains at least one root of the polynomial \(P\), say \(\chi_1\). Let us place the remaining roots of the polynomial \(P\) lying in the upper half-
in order of proximity, \(x_2, x_3,\ldots,x\), so that
\[ |x_1-x_2|\leq |x_1-x_3|\leq \cdots \leq |x_1-x_k| \qquad \left(k\leq \frac n2\right). \]
Now put \(|x_1-x_i|=h^{-\rho_i}\) \((i=2,3,\ldots,k)\). Take an arbitrary, but subsequently fixed, \(\varepsilon>0\), put \(m=[n/\varepsilon]+1\), and define the integers \(r_i\) by the inequalities
\[ \frac{r_i}{m}\leq \rho_i<\frac{r_i+1}{m}\qquad (i=2,3,\ldots,k). \]
Then
\[ h^{1-s-\varepsilon}\ll |P'(x_1)|\ll h^{1-s},\qquad h^{-\frac{r_i+1}{m}}< |x_1-x_i|\leq h^{-\frac{r_i}{m}} \qquad (i=2,3,\ldots,k), \]
where \(s=\frac1m(r_1+r_2+\cdots+r_k)\). It is easy to see that there exist at most \(c(n,\varepsilon)\) different systems \((r_1,r_2,\ldots,r_k)\). Next, we divide the polynomials \(P\) with the same number \(k\) into three classes depending on the three possibilities:
\[ 1^\circ.\quad \frac{r_2}{m}<\frac n4-\frac{s_1}{2}, \]
\[ 2^\circ.\quad s_1<\frac n6,\quad \frac{r_2}{m}\geq \frac n4-\frac{s_1}{2}, \]
\[ 3^\circ.\quad s_1\geq \frac n6. \]
Here \(s_1=s-r_2/m\). We now consider separately the polynomials of each class.
\(1^\circ.\) Let \(\omega_1\) be the point on the boundary \(\sigma_1(P)\) nearest to \(x_1\). We have
\[ |\omega_1-x_1|\ll h^{-\frac12(w_0+1-s-\varepsilon)}. \]
Therefore \(|\omega_1-x_1|\leq |x_1-x_2|\), so that
\[ |\omega_1-x_1|\gg |P(\omega_1)|:|P'(x_1)|>h^{-w_0-1+s}. \]
Since the circle of radius \(|\omega_1-x_1|\) with center at \(x_1\) lies inside \(\sigma_1(P)\), it follows that
\[ \operatorname{mes}\sigma_1(P)\gg h^{-2(w_0+1-s)}, \]
and the number of polynomials of the first class of the given height \(h\) will be
\[ \ll h^{2(w_0+1-s)}. \]
Consequently, the measure of the set of numbers \(\omega\) for which (1) holds with \(P\in \mathfrak P_n(h)\) of the first class will be
\[ \ll h^{-2(w+1-s-\varepsilon)}h^{2(w_0+1-s)}=h^{-2(w-w_0)+2\varepsilon}. \]
If \(w>w_0+\frac12+\varepsilon=w_{n-1}+\frac12+\delta+\varepsilon\), then inequality (1) has a finite number of solutions in polynomials of the first class for almost all \(\omega\).
\(2^\circ.\) In this case \(r_3/m\leq n/4-s_1/2\). Suppose that there exists a pair of polynomials \(P_1,P_2\in\mathfrak P_n(h)\) satisfying
\[ |x_1^{(1)}-x_1^{(2)}|<h^{-\frac n4+\frac{s_1}{2}}. \]
Since then
\[ |x_i^{(1)}-x_j^{(2)}|\leq 2h^{-\frac1m r_{\max(i,j)}}+h^{-\frac n4+\frac{s_1}{2}} \qquad (i,j=1,2,\ldots,k), \]
we have
\[ |x_i^{(1)}-x_j^{(2)}|\ll \begin{cases} h^{-\frac n4+\frac{s_1}{2}}, & \text{if } \max(i,j)\leq 2,\\[4pt] h^{-\frac1m r_{\max(i,j)}}, & \text{if } \max(i,j)\geq 3. \end{cases} \]
Consequently,
\[ h \ll |R(P_1,P_2)| \ll h^{2n} h^{-8\left(\frac n4-\frac{s_1}{2}\right)} \prod_{\max(i,j)\ge 3} h^{-\frac{2}{m} r\max(i,j)} \ll h^{-6s_1}, \]
which is impossible. Thus, the circle with center at \(\chi_1^{(1)}\) and radius \(h^{-\frac n4+\frac{s_1}{2}}\) is always free of other roots \(\chi_1^{(2)}\). Therefore the number of polynomials of the second class is
\[ \ll h^{\frac n2-s_1}. \]
The measure of the set of numbers \(\omega\) for which (1) holds with polynomials \(P \in \mathfrak P_n(h)\) of the second class will be
\[ \ll h^{-w-1+s_1+\varepsilon} h^{\frac n2-s_1} = h^{-w-1+\frac n2+\varepsilon}, \]
and one should take \(w>n/2+\varepsilon\).
\(3^\circ\). Let, as in \(1^\circ\), \(\omega_1\) be the point on the boundary \(\sigma_1(P)\) nearest to \(\chi_1\). First consider those polynomials for which
\[
|\omega_1-\chi_1| \ge 2|\chi_1-\chi_2|.
\]
We have
\[
|\omega_1-\chi_1| \gg h^{-1/2(w_0+1-s_1)},
\]
and the number of polynomials is
\[
\ll h^{w_0+1-s_1}.
\]
By Lemma 1, the measure of the set of numbers \(\omega\) for which (1) holds with \(P \in \mathfrak P_n(h)\) of the third class will be
\[ \ll h^{5/3-\frac{4w-n}{3}} h^{w_0+1-s_1} \ll h^{-2/3-4/3\,w+w_0+\frac n6}. \]
Therefore we assume
\[
4w>3w_0+\frac n2+1.
\]
The remaining polynomials of the third class satisfying
\[
|\omega_1-\chi_1| \le 2|\chi_1-\chi_2|
\]
are considered analogously to the polynomials of the first class.
Summing up, we conclude from the preceding that
\[ w_n \ll \max\left(w_{n-1}+\frac12,\ \frac n2,\ \frac34 w_{n-1}+\frac n8+\frac14\right). \]
In view of (4), it follows from this that
\[
w_n \ll n/2 \qquad (n=1,2,\ldots).
\]
We note that the above arguments concerning polynomials of the first and third classes do not prevent an attempt to obtain the equalities
\[
\theta_n=1,\qquad \eta_n=\frac12-\frac{1}{2n}
\]
for all \(n\). By developing somewhat the reasoning concerning polynomials of the second class, one can obtain inequalities of the form
\[ \theta_n \ll 1+\frac{c_1}{n}, \qquad \eta_n \ll \frac12-\frac{c_2}{2n} \qquad (n=1,2,\ldots) \]
with some absolute constants \(c_1,c_2\), \(0<c_1,c_2<1\).
Institute of Mathematics and Computer Technology
Academy of Sciences of the BSSR
Received
16 IX 1963
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