UDC 511

MATHEMATICS

V. G. SPRINDŽUK

ASYMPTOTICS OF THE NUMBER OF SOLUTIONS OF CERTAIN DIOPHANTINE INEQUALITIES

(Presented by Academician I. M. Vinogradov on 30 V 1966)

Let \(\alpha,\ \beta\) be real numbers; let \(\chi(q)\) be a positive monotonically decreasing function defined for natural numbers \(q\); let \(\|x\|=\min(\{x\},1-\{x\})\) be the distance from \(x\) to the nearest integer. It is known that for almost all \(\alpha\) the number of solutions of the inequality

\[ \|\alpha q+\beta\|<\chi(q) \tag{1} \]

in integers \(q,\ 1\le q\le Q\), is asymptotically equal to \(2\Psi(Q)+O(\Psi^{1/2+\varepsilon}(Q))\), where \(\Psi(Q)=\sum_{q=1}^{Q}\chi(q)\), if \(\Psi(Q)\to\infty\) (see, for example, \((^5,^8,^11,^12)\)). It is of interest to give “individual” variants of this fact, i.e. to indicate concrete numbers \(\alpha\) (or classes of numbers \(\alpha\)) for which the analogous assertion is true. Recently S. Lang \((^6,^7)\) proved for quadratic irrationalities \(\alpha\) and functions \(\chi(q)\) decreasing as \(q^{-1}\) or considerably more slowly than \(q^{-1}\), that the number of solutions of (1) with \(\beta=0\) has the asymptotic \(2\Psi(Q)\). Lang’s arguments are based on the arithmetic of quadratic fields and do not admit any generalization to the case of other \(\alpha\).

In this note we show, in particular, that if \(\alpha\) satisfies the following condition (A): the number of solutions of the inequality \(\|\alpha q\|<q^{-1-\varepsilon}\) is finite for every \(\varepsilon>0\), then for any \(\beta\) and any \(\sigma,\ 0<\sigma<1\), the number of solutions of the inequality

\[ \|\alpha q+\beta\|<q^{-\sigma},\qquad 1\le q\le Q, \tag{2} \]

is asymptotically equal to

\[ \frac{2}{1-\sigma}Q^{1-\sigma}+O\left(Q^{(1-\sigma)/2+\varepsilon}\right). \]

The following more general result is true.

Theorem. Let \(\chi(t)\) be a positive continuously differentiable function defined for \(t\ge1\), whose derivative satisfies the conditions: 1) \(\chi'(t)<0\) for all \(t\ge1\), \(|\chi'(t)|\) tends monotonically to zero as \(t\to\infty\); 2) \(|\chi'(t)|\gg t^{-2}\) \((t\gg1)\), and if \(t<t_1\le t\), then \(|\chi'(t)|\ll|\chi'(t_1)|\). Then, for \(\alpha\) satisfying condition (A), and for any \(\beta\), the number of solutions of inequality (1) is asymptotically equal to

\[ 2\Psi(Q)+O\left(\Psi(Q)\chi^{-1}(Q)|\chi'(Q)|^{1/2}Q^{\varepsilon}\right) +O\left(\chi(Q)|\chi'(Q)|^{-1/2}\right), \]

where

\[ \Psi(Q)=\int_{1}^{Q}\chi(t)\,dt. \]

This theorem is an immediate consequence of the well-known fact of the strong uniform distribution of the fractional parts \(\|\alpha q+\beta\|\) for \(\alpha\) satisfying condition (A) (see Lemma 1), and of Lemma 2 below, which makes it possible to determine the asymptotics of the number of solutions of the inequality \(a_q<\chi(q)\), if it is known that the sequence \(a_q\) is well distributed on some interval \((0,\lambda)\), \(\lambda>0\).

It is known that condition (A) is satisfied by irrational algebraic numbers \((^{10},{}^{2})\), and also by numbers of the form \(e^r\) \((r \ne 0)\) and \(\ln r\) \((r \ne 0,1)\), where \(r\) is a rational number \((^{3},{}^{4},{}^{9})\).

Lemma 1. Let \(\alpha\) be a real number satisfying condition (A); let \(\beta\) be any real number; let \(I\) be an arbitrary interval inside the interval \((0,\tfrac12)\); and let \(N_Q(I)\) be the number of fractional parts \(\| \alpha q+\beta \|\), \(1 \le q \le Q\), lying in \(I\). Then
\[ N_Q(I)=2|I|Q+O(Q^\varepsilon), \]
where \(|I|\) is the length of the interval \(I\), and the estimate is uniform over all intervals \(I\).

The proof of this lemma is obtained without difficulty by applying I. M. Vinogradov’s “little cups” method \(((^{1}),\) p. 260).

Lemma 2. Let \(a_q\) \((q=1,2,\ldots)\) be a sequence of real numbers lying in the interval \((0,\lambda)\), where \(\lambda>0\) is some fixed number; let \(I\) be an arbitrary interval inside the interval \((0,\lambda)\); let \(N_Q(I)\) be the number of those \(a_q\), \(1 \le q \le Q\), which lie in \(I\), and suppose that
\[ N_Q(I)=\lambda^{-1}|I|Q+R_Q(I),\qquad |R_Q(I)|\le R(Q), \tag{3} \]
where the quantity \(R(Q)\) does not depend on \(I\), and \(R(Q)\) increases monotonically to infinity. Finally, let \(\chi(t)\) be a function satisfying the conditions of the theorem. Then the number \(N(Q)\) of those \(a_q\) which satisfy the inequality \(a_q<\chi(q)\), \(1\le q\le Q\), is equal to
\[ \lambda^{-1}\Psi(Q)+L(Q), \]
\[ L(Q)=O\!\left(\Psi(Q)\chi^{-1}(Q)|\chi'(Q)|^{1/2}R(Q)+\chi(Q)|\chi'(Q)|^{-1/2}\right). \]

Proof. Define the numbers \(p_k\) from the conditions \(p_1=1\),
\[ p_{k+1}-p_k=|\chi'(p_k)|^{-1/2}. \]
It is clear that the numbers \(p_k\) are positive and the difference \(p_{k+1}-p_k\) increases monotonically. Denote by \(N_0(k)\) the number of those \(a_q\) for which the inequality \(a_q<\chi(q)\) holds under the condition \(p_k\le q<p_{k+1}\); by \(N_1(k)\), the number of those \(a_q\) for which \(a_q<\chi(p_k)\) under \(p_k\le q\le p_{k+1}\); and, analogously, denote by \(N_2(k)\) the number of those \(a_q\) for which \(a_q<\chi(p_{k+1})\) under \(p_k\le q<p_{k+1}\). Since the function \(\chi(t)\) decreases monotonically, we have \(N_2(k)\le N_0(k)\le N_1(k)\), i.e.
\[ N_0(k)=N_1(k)-\theta\bigl(N_1(k)-N_2(k)\bigr),\qquad 0\le\theta\le 1. \]
Since the difference \(N_1(k)-N_2(k)\) is the number of those \(a_q\) which fall into the interval \((\chi(p_{k+1}),\chi(p_k))\), by the condition of the lemma (3) this difference does not exceed the quantity
\[ O\!\left((\chi(p_k)-\chi(p_{k+1}))(p_{k+1}-p_k)+R(p_{k+1})\right). \]

Define \(k_0\) from the condition \(p_{k_0}\le Q<p_{k_0+1}\). Then, since
\[ N_1(k)=\lambda^{-1}\chi(p_k)(p_{k+1}-p_k)+O(R(p_{k+1})), \]
we have
\[ N(Q)=\sum_{k=1}^{k_0-1}N_1(k)+L_1(Q)= \]
\[ =\sum_{k=1}^{k_0-1}\lambda^{-1}\chi(p_k)(p_{k+1}-p_k)+L_2(Q) =\lambda^{-1}\Psi(Q)+L_3(Q), \]
where all quantities \(L_j(Q)\) have order
\[ O\!\left(\sum_{k=1}^{k_0-1}(\chi(p_k)-\chi(p_{k+1}))(p_{k+1}-p_k)+k_0R(Q)+\chi(p_{k_0})(Q-p_{k_0})\right). \tag{4} \]

Let \(x,y\) be real numbers satisfying \(p_k\le x<y\le p_{k+1}\). By the choice of the numbers \(p_k\) and the monotone variation of the functions \(\chi(t)\), \(|\chi'(t)|\), we find
\[ 0<(\chi(x)-\chi(y))(y-x)\le |\chi'(x)|(y-x)^2\le \]
\[ \le |\chi'(p_k)|(p_{k+1}-p_k)(y-x)=(y-x)(p_{k+1}-p_k)^{-1}\le 1. \]
Therefore
\[ \chi(p_{k_0})(Q-p_{k_0})\le \chi(Q)(Q-p_{k_0})+1\le \chi(Q)|\chi'(Q)|^{-1/2}+1 \]
and the remainder term (4) will be a quantity
\[ O\!\left(k_0R(Q)+\chi(Q)|\chi'(Q)|^{-1/2}\right). \]

Let us estimate the quantity \(k_0\). Setting \(k_1=[k_0/2]\), we have

\[ \int_{q_{k_1}}^{q_{k_0}} \chi(t)\,dt \geq \chi(Q)(p_{k_0}-p_{k_1}) =\chi(Q)\sum_{k_1\leq k<k_0}(p_{k+1}-p_k) = \]

\[ =\chi(Q)\sum_{k_1\leq k<k_0}\left|\chi'(p_k)\right|^{-1/2} \geq \chi(Q)\left|\chi'(p_{k_1})\right|^{-1/2}(k_0-k_1). \]

By virtue of the conditions imposed on \(\left|\chi'(t)\right|\), we find

\[ \frac{p_{k+1}}{p_k} = 1+p_k^{-1}\left|\chi'(p_k)\right|^{-1/2}\ll 1, \qquad p_{k+1}\ll p_k; \]

therefore \(Q\ll p_{k_1}\), and then \(\left|\chi'(p_{k_1})\right|\ll \left|\chi'(Q)\right|\). Thus,

\[ k_0\ll \chi^{-1}(Q)\left|\chi'(Q)\right|^{1/2}\int_1^Q \chi(t)\,dt, \]

which completes the proof of Lemma 2.

In the case of simultaneous approximations

\[ \max\left(\|a_1q+\beta_1\|,\ldots,\|a_nq+\beta_n\|\right)<\chi(q) \tag{5} \]

analogous results hold, as is not difficult to establish by only slightly developing the arguments. Condition (A) here must be replaced by the condition: the number of solutions of the inequality

\[ \|a_1q\|\cdot \|a_2q\|\ldots \|a_nq\|<q^{-1-\varepsilon} \]

in integers \(q>0\), for every \(\varepsilon>0\), is finite; or, equivalently, the number of solutions of the inequality

\[ \|a_1a_1+\alpha_2a_2+\ldots+a_na_n\|>(a_1'a_2'\ldots a_n')^{-n-\varepsilon} \]

in integers \(a_i\) \((i=1,2,\ldots,n)\), where \(a_i'=\max(1,|a_i|)\), is finite for every \(\varepsilon>0\). For example, one may take \(\alpha_i=e^{r_i}\), where \(r_i\ne0\) are pairwise distinct rational numbers \((^4)\). Under these conditions, the number of solutions of the system of inequalities (5) with \(\chi(q)=q^{-\sigma}\), \(0<\sigma<1/n\), \(1\leq q\leq Q\), is asymptotically equal to

\[ \frac{2^n}{1-n\sigma}Q^{1-n\sigma} + O\left(Q^{\frac{1-n\sigma}{2}+\varepsilon}\right). \]

Institute of Mathematics
Academy of Sciences of the BSSR

Received
16 V 1966

REFERENCES

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