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MATHEMATICS
Yu. V. KASHIRSKII
ON TRANSFERENCE THEOREMS
(Presented by Academician I. M. Vinogradov on 4 XI 1962)
Let \(\bar{x}=\max(1,|x|)\), and let \(\langle x\rangle\) denote the distance from \(x\) to the nearest integer. Consider the linear forms
\[ L_j(x_1,\ldots,x_m)=\sum_{i=1}^{m} a_{ij}x_i;\qquad M_i(u_1,\ldots,u_n)=\sum_{j=1}^{n} a_{ij}u_j . \]
From the classical transference theorems of A. Ya. Khinchin \((^{1,2})\) it follows that, if there is no nontrivial integral solution of the inequalities
\[ \max \langle M_i(u_1,\ldots,u_n)\rangle \leq D;\qquad \max |u_j|\leq Y \quad (i=1,\ldots,m;\ j=1,\ldots,n), \tag{1} \]
then there is also no nontrivial integral solution for the inequalities
\[ \max \langle L_j(x_1,\ldots,x_m)\rangle \leq C;\qquad \max |x_i|\leq X \quad (i=1,\ldots,m;\ j=1,\ldots,n), \]
where
\[ X=C_1(m,n)\frac{Y^n}{D^{1-m}};\qquad C=C_2(m,n)\frac{D^m}{Y^{1-n}} \]
(see, for example, Theorem 2, Ch. 5 in \((^{3})\)).
Transference theorems in which, as above, we deal with maxima of linear forms and variables will be called cubic. We shall consider transference theorems of another kind, where conditions are imposed on products of linear forms and variables. In particular, conditions (1) are replaced by conditions of the form
\[ (\bar{x}_1\ldots \bar{x}_n)^{\gamma} [\ln(\bar{x}_1+1)\ldots \ln(\bar{x}_n+1)]^{\beta} \prod_{i=1}^{m}\langle M_i(x_1,\ldots,x_n)\rangle \geq C_0 . \]
We shall call these the hyperbolic transference theorems. A special case of theorems of this type was considered by A. Ya. Khinchin \((^{2})\) and later by Dyson \((^{4})\). Usually the proofs of various transference theorems are based on the application of Dirichlet’s principle (an exception is the analytic theorem of A. O. Gelfond \((^{3})\)). In the present paper, in contrast to this, another method of proof is used, proposed by N. M. Korobov. It is based on the estimation of trigonometric sums.
§ 1. Consider the inequalities
\[ \langle a_1x_1+\cdots+a_nx_n\rangle \geq \frac{C}{\bar{x}_1\ldots \bar{x}_n \ln^{\gamma}(\bar{x}_1+1)\ldots \ln^{\gamma}(\bar{x}_n+1)}, \tag{2} \]
\[ \langle a_1x\rangle\cdots \langle a_nx\rangle \geq \frac{C_1}{x\ln^{\gamma_1}(x+1)}, \tag{3} \]
where \(C=C(a_i,n)\), \(C_1=C_1(a_i,n)\).
Theorem 1. If inequality (2) is satisfied for any integers \((x_1,\ldots,x_n)\neq(0,\ldots,0)\), then for every integer \(x>0\), with \(\gamma_1=(\gamma+2)n^2\), inequality (3) is also satisfied. The converse theorem is also true with \(\gamma=(\gamma_1+1+n)n\).
We shall confine ourselves to proving only the second part of the theorem. The proof is based on the following
Lemma. If for \(m>0\) the inequality
\[ \langle\!\langle \alpha_1 m\rangle\!\rangle \cdots \langle\!\langle \alpha_s\rangle\!\rangle \ge \frac{C_2}{m\ln^\gamma(m+1)}, \]
holds, then
\[ \sum_{k=1}^{m} \frac{1}{\langle\!\langle \alpha_1 k\rangle\!\rangle \cdots \langle\!\langle \alpha_s k\rangle\!\rangle} \le C_3 m \ln^{\gamma+s}(m+1). \]
This assertion is obtained with the aid of Abel’s transformation and the application of Dirichlet’s principle.
Let us pass to the proof of the theorem. Denote by \(N_{p_1,\ldots,p_n}(0,\alpha)\) the number of incidences of \(\{\alpha_1x_1+\cdots+\alpha_nx_n\}\) in \((0,\alpha)\) for \(1\le x_1\le p_1,\ldots,1\le x_n\le p_n\). Introduce for the interval \((0,\alpha)\) functions \(\psi(x)\) and \(\psi_1(x)\), making it possible to approximate \(N_{p_1,\ldots,p_n}(0,\alpha)\) from above and below (see \((^5)\), p. 260). Then
\[ \sum_{x_1=1}^{p_1}\cdots\sum_{x_n=1}^{p_n} \psi_1(\alpha_1x_1+\cdots+\alpha_nx_n) \le N_{p_1,\ldots,p_n}(0,\alpha) \le \]
\[ \le \sum_{x_1=1}^{p_1}\cdots\sum_{x_n=1}^{p_n} \psi(\alpha_1x_1+\cdots+\alpha_nx_n). \]
Hence we obtain that
\[ R= \left|N_{p_1,\ldots,p_n}(0,\alpha)-p_1\cdots p_n\alpha\right| \le \]
\[ \le 2\left| \sum_{m=1}^{\infty} C(m) \sum_{x_1,\ldots,x_n} \exp\left[2\pi i\left(\sum_{i=1}^{n}\alpha_i x_i\right)m\right] \right| +2p_1\cdots p_n\delta, \]
where
\[ |C(m)|< \begin{cases} \dfrac{1}{\pi m}, & \text{for } \delta<\dfrac{1}{\pi m},\\[6pt] \dfrac{1}{\pi^2m^2\delta}, & \text{for } \delta\ge\dfrac{1}{\pi m}. \end{cases} \]
Choose
\[ \delta=\frac{r_1}{\pi p_1\cdots p_n} \]
and split the interval of summation over \(m\)
\[ R\le C_4\left( \sum_{m=1}^{p_1\cdots p_n} \frac{1}{m\langle\!\langle\alpha_1m\rangle\!\rangle\cdots\langle\!\langle\alpha_nm\rangle\!\rangle} + \sum_{m=p_1\cdots p_n+1}^{\infty} \frac{p_1\cdots p_n}{m^2\langle\!\langle\alpha_1m\rangle\!\rangle\cdots\langle\!\langle\alpha_nm\rangle\!\rangle} \right). \]
Using Abel’s transformation and applying the lemma, we obtain for the first sum
\[ |S_1|\le C_5\left( \frac{1}{p_1\cdots p_n} \sum_{m=1}^{p_1\cdots p_n} \frac{1}{\langle\!\langle\alpha_1m\rangle\!\rangle\cdots\langle\!\langle\alpha_nm\rangle\!\rangle} + \sum_{m=1}^{p_1\cdots p_n} \frac{1}{m^2} \sum_{k=1}^{m} \frac{1}{\langle\!\langle\alpha_1k\rangle\!\rangle\cdots\langle\!\langle\alpha_nk\rangle\!\rangle} \right) \le \]
\[ \le C_6\left[ \ln^{\gamma_1+n}(p_1\cdots p_n+1) + \sum_{m=1}^{p_1\cdots p_n} \frac{\ln^{\gamma_1+n}(m+1)}{m} \right] \le \]
\[ \le 2C_6\ln^{\gamma_1+n+1}(p_1\cdots p_n+1). \]
Similarly,
\[ |S_2|\le C_7\ln^{\gamma_1+n}(p_1\cdots p_n+1), \]
\[ R\le C_8\ln^{\gamma_1+n+1}(p_1\cdots p_n+1). \tag{4} \]
We shall show that if estimate (4) holds for \(p_1,\ldots,p_n\), then for
\(x_i=\pm [p_i/R]\) the inequality
\[ \{a_1x_1+\cdots+a_nx_n\}\leq \left\langle {k\over 4p_1\cdots p_n}\right\rangle ={C_9\over R^n\bar x_1\cdots \bar x_n} \tag{5} \]
will follow.
Indeed, suppose
\[ \{a_1x_1+\cdots+a_nx_n\}\leq {1\over 4p_1\cdots p_n}. \]
Choose \(T=2R,\ \alpha=R/p_1\cdots p_n\). Then, evidently,
\[ \{k(a_1x_1+\cdots+a_nx_n)\}\leq {k\over 4p_1\cdots p_n}<\alpha \quad\text{for } k=1,\ldots,T . \]
Thus, on the interval \((0,\alpha)\) there lie at least \(T\) fractional parts. Consequently,
\[ T\leq N_{p_1,\ldots,p_n}(0,\alpha)<\alpha p_1\cdots p_n+R=2R=T. \]
The contradiction obtained proves inequality (5). It is easy to see that from (5) the estimate
\[ \{a_1x_1+\cdots+a_nx_n\}\geq {C_{10}\over \bar x_1\cdots \bar x_n \ln^{(v_1+1+n)n}(\bar x_1\cdots \bar x_n+1)} \tag{6} \]
follows, where, as above, \(x_i=\pm [p_i/R]\).
When \(p_1,\ldots,p_n\) run through all natural numbers, the collection
\(x_1,\ldots,x_n\) assumes all integer values. By the same method one obtains the inequality
\[ 1-\{a_1x_1+\cdots+a_nx_n\}\geq {C_{10}\over \bar x_1\cdots \bar x_n \ln^{(v_1+1+n)n}(\bar x_1\cdots \bar x_n+1)} . \tag{7} \]
From (6) and (7) the assertion of the theorem follows.
Remark 1. Analogously one obtains transference theorems relating a system of linear forms to the transposed system.
Remark 2. With the aid of A. O. Gel’fond’s theorem \((^3)\), similar theorems are obtained, but with less precise estimates.
2. Let \(p\) be an arbitrary integer greater than zero. Consider the systems of congruences
\[ \begin{aligned} a_{11}x_1+\cdots+a_{1n}x_n &\equiv x_{n+1},\\ &\ldots\\ a_{m1}x_1+\cdots+a_{mn}x_n &\equiv x_{n+m}, \end{aligned} \left\}\pmod p,\right. \tag{8} \]
\[ \begin{aligned} a_{11}x_1+\cdots+a_{m1}x_m &\equiv x_{m+1},\\ &\ldots\\ a_{1n}x_1+\cdots+a_{mn}x_m &\equiv x_{m+n}, \end{aligned} \left\}\pmod p,\right. \tag{9} \]
and let \(q\) and \(Q\) be the minimal values of \(\sum_{i=1}^{n+m}x_i\), where
\(x_1,\ldots,x_{n+m}\) are respectively solutions of systems (8) and (9).
Theorem 2. There exists the following dependence between \(q\) and \(Q\):
\[ q\geq {cQ^{n+m-1}\over p^{(n-1)(n+m)}\ln p^{(n+m)(n+m-1)}}, \qquad Q\geq {c_1q^{n+m-1}\over p^{(m-1)(n+m)}\ln p^{(n+m)(n+m-1)}}. \]
Analogous cubic transference theorems are also valid for a system of comparisons.
Using the results of the transference theorems for comparisons, one can prove the following theorem, which somewhat extends the classical transference theorems indicated in the introduction.
Theorem 3. Let the inequalities
\[ \max \langle M_i(u_1,\ldots,u_m)\rangle \leqslant D,\quad |u_j|\leqslant Y \]
\[ (i=1,2,\ldots,n;\ j=1,2,\ldots,m) \]
have no solutions except the trivial one. Then the inequality
\[ \prod_{i=1}^{n}\overline{x_i}\prod_{j=1}^{m}\langle L_j(x_1,\ldots,x_n)\rangle \leqslant C(m,n)\, \frac{(u^{m}D^{n})^{\,n+m-1}} {\ln^{(n+m)(n+m-1)}\!\left(\dfrac{XY}{DC}\right)}, \]
has no nontrivial solutions, where
\[ X=\max_{i=1,\ldots,n}|x_i|,\quad C=\min_{j=1,\ldots,m}\langle L_j x_1,\ldots,x_n\rangle . \]
In conclusion I express my deep gratitude to A. O. Gelfond and N. M. Korobov for posing the problem and for valuable advice and suggestions.
Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR
Received
22 X 1962
References
- A. Khintschine, Rend. Circ. Mat. Palermo, 50, 170 (1926).
- A. Khintschine, Math. Zs., 22, 274 (1925).
- J. W. S. Cassels, An Introduction to Diophantine Approximation, Moscow, 1961.
- F. J. Dyson, Proc. Lond. Math. Soc., ser. 2, 49, 6, 409 (1947).
- I. M. Vinogradov, Selected Works, Moscow, 1952.