Full Text
V. G. Sprindzhuk
More on Mahler’s Hypothesis
(Presented by Academician I. M. Vinogradov, 16 XI 1963)
In the preceding note \((^1)\) on the same topic, a scheme was set forth for proving K. Mahler’s hypothesis on the measure of the set of complex \(S\)-numbers:
\[ \sup_{(n)} \eta_n=\frac{1}{2}\qquad (n=1,2,\ldots). \]
In this note we show how a certain modification of the previous arguments leads to a proof of the real case of K. Mahler’s hypothesis:
\[ \sup_{(n)} \theta_n=1\qquad (n=1,2,\ldots). \tag{1} \]
Let \(\mathfrak P_n(h)\) be the set of irreducible primitive polynomials with integer coefficients
\[ P=a_0+a_1x+\cdots+a_nx^n \]
satisfying \(n\geqslant 3\),
\[ \max\left(|a_0|,|a_1|,\ldots,|a_{n-1}|\right)\leqslant a_n=h. \tag{2} \]
Denote by \(x_1,x_2,\ldots,x_n\) the roots of the polynomial \(P\), so that
\[ P(x)=h(x-x_1)\cdots(x-x_n). \]
Lemma 1. Let \(P\in\mathfrak P_n(h)\), let \(\omega\) be a real number, \(|P(\omega)|<h^{-w}\), \(w>0\). Put
\[ |\omega-x_1|=\min_{(i)}|\omega-x_i|\qquad (i=1,2,\ldots,n). \]
Then
\[ |\omega-x_1|<c(n)h^{-1-(2w-n)/3}|D(P)|^{-1/6}. \]
Lemma 2. Suppose that, under the conditions of the preceding lemma, \(w\geqslant n-1\), and \(x_2\) is the root of the polynomial \(P\) nearest to \(x_1\).
Then
\[ \omega-x_1 \asymp \begin{cases} |P(\omega)|:P'(x_1), & \text{if } |\omega-x_1|\leqslant 2|x_1-x_2|,\\[4pt] \left(|P(\omega)|\,|x_1-x_2|:|P'(x_1)|\right)^{1/2}, & \text{if } |\omega-x_1|>2|x_1-x_2|. \end{cases} \]
Lemma 3. Let \(\Delta\) be a measurable set on the line, \(\operatorname{mes}\Delta<\varepsilon\), and let a system
\[ \Lambda=\bigcup_{i=1}^{\infty}\lambda_i \]
of intervals \(\lambda_i\) be given with the conditions
\[ \operatorname{mes}(\lambda_i\cap\Delta)\geqslant \frac12\operatorname{mes}\lambda_i\qquad (i=1,2,\ldots). \]
Then
\[ \operatorname{mes}\Lambda<4\varepsilon. \]
We proceed to the proof of (1). Let \(w_n=n\theta_n\) \((n=1,2,\ldots)\). Obviously, we may assume that \(|\omega|<c\). Let \(\sigma(P)\) be a system of pairwise nonintersecting intervals \(\sigma_i(P)\), in which the inequality
\[ |P(\omega)|<h^{-w},\qquad w=w_{n-1}+\delta, \tag{3} \]
is satisfied.
where $\delta>0$ is an arbitrary small number, but fixed in what follows; $\sigma(P)=\sum\sigma_i(P)$. Arguing as in the case of complex numbers, on the basis of Lemma 3 we conclude that only those intervals $\sigma_i(P)$ are of interest in which the set of points $\omega$ belonging to other systems $\sigma(Q)$, $Q\in\mathfrak P_n(h)$, has measure not less than $\frac12\operatorname{mes}\sigma_i(P)$.
Let $\sigma_1(P)$ be one of the intervals of the system $\sigma(P)$. Define the root $\chi_1$ of the polynomial $P$ for which in $\sigma_1(P)$ there is at least one point $\omega$ satisfying the condition
\[ |\omega-\chi_1|=\min|\omega-\chi_i|\qquad (i=1,2,\ldots,n). \]
Arrange the remaining roots of the polynomial $P$ in the order $\chi_2,\chi_3,\ldots,\chi_n$ so that
\[ |\chi_1-\chi_2|\leq |\chi_1-\chi_3|\leq \cdots \leq |\chi_1-\chi_n|. \]
Put $|\chi_1-\chi_i|=h^{-\rho_i}$ $(i=2,3,\ldots,n)$. Take an arbitrary, but fixed in what follows, $\varepsilon>0$, put $m=[n/\varepsilon]+1$, and define the integers $r_i$ by the inequalities
\[ \frac{r_i}{m}\leq \rho_i<\frac{r_i+1}{m}\qquad (i=2,3,\ldots,k). \tag{4} \]
Then there exist no more than $c(n,\varepsilon)$ distinct systems $(r_2,r_3,\ldots,r_n)$ generated by polynomials $P\in\mathfrak P_n(h)$. If there exist infinitely many polynomials $P$ satisfying condition (4), then
\[ \sum_{j=2}^{n}(j-1)\frac{r_j}{m}\leq n-1, \tag{5} \]
as follows from considering the discriminant $D(P)$ of the polynomial $P$ in the form
\[ D(P)=h^{2n-2}\prod_{1\leq i<j\leq n}(\chi_i-\chi_j)^2. \]
Polynomials $P$ having the same systems $(r_2,r_3,\ldots,r_n)$ are assigned to the first or the second class depending on whether the following conditions are fulfilled:
1) $\displaystyle \frac{r_2}{m}<\frac{n-s_1}{2};$
2) $\displaystyle \frac{r_2}{m}\geq \frac{n-s_1}{2},$
where $s_1=\frac1m(r_3+r_4+\cdots+r_n)$. Polynomials of the first class are considered in the same way as in the preceding note. In the case of polynomials of the second class, however, by virtue of (5), we have $r_2/m+2s_1\leq n-1$. Therefore
\[ \frac{n-s_1}{2}+2s_1\leq \frac{r_2}{m}+2s_1\leq n-1,\qquad s_1\leq \frac{n-2}{3}. \]
Further, $2r_3/m+s_1\leq 3s_1\leq n-2$, so that $r_3/m\leq (n-s_1)/2-1$. Consequently,
\[ \frac{r_2}{m}\geq \frac{n-s_1}{2}>\frac{r_3}{m}. \tag{6} \]
Now let $\mathfrak P_n^k$ be the set of polynomials $P\in\bigcup_{h=1}^{\infty}\mathfrak P_n(h)$ satisfying the condition $2^{k-1}<h(P)\leq 2^k=H$, where $h(P)$ is the height of $P$. Suppose that there exists a pair of polynomials $P_1,P_2\in\mathfrak P_n^k$ satisfying $|\chi_1^{(1)}-\chi_1^{(2)}|<cH^{-(n-s_1)/2}$, where $\chi_1^{(1)}$, $\chi_1^{(2)}$ are, respectively, the roots of the polynomials $P_1,P_2$ relative to which are defi-
the membership of these polynomials in the second class has been determined. Then
\[ \left| \chi_i^{(1)}-\chi_j^{(2)} \right| \ll 2H^{-\frac{1}{m} r_{\max(i,j)}}+cH^{-(n-s_1)/2}. \]
In view of (6), we now find
\[ \left| \chi_i^{(1)}-\chi_j^{(2)} \right| \ll \begin{cases} cH^{-(n-s_1)/2}, & \text{if } \max(i,j)\leqslant 2,\\ H^{-\frac{1}{m} r_{\max(i,j)}}, & \text{if } \max(i,j)\geqslant 3. \end{cases} \]
Therefore
\[ \begin{aligned} 1 \leqslant |R(P_1,P_2)| &\leqslant (h_1h_2)^n \prod_{1\leqslant i,j\leqslant n} \left| \chi_i^{(1)}-\chi_j^{(2)} \right| \\ &\ll H^{2n}c^4H^{-4(n-s_1)/2} \prod_{\max(i,j)\geqslant 3} H^{-\frac{1}{m} r_{\max(i,j)}} \leqslant c^4 H^{2s_1-5s_1} \leqslant c^4 . \end{aligned} \]
For sufficiently small \(c>0\), the inequality obtained is impossible. Consequently, the number of polynomials \(P\) of the second class in \(\mathfrak{M}_n^k\) will be \(\ll H^{(n-s_1)/2}\). This allows us to conclude that
\[ w_n \leqslant \max(w_{n-1}+1,\ n-1) \qquad (n=3,4,\ldots). \]
Since \(w_2=2\) and \(w_n\geqslant n\), it must be that \(w_n=n\) \((n=2,3,\ldots)\).
I express my sincere gratitude to Prof. Yu. V. Linnik, who showed great interest in the author’s work.
Institute of Mathematics and Computer Technology
Academy of Sciences of the BSSR
Received
31 X 1963
REFERENCES
- V. G. Sprindzhuk, DAN, 154, No. 4 (1964).