MATHEMATICS

L. P. USOLTSEV

ON AN EXPONENTIAL RATIONAL TRIGONOMETRIC SUM OF A SPECIAL FORM

(Presented by Academician I. M. Vinogradov on 24 I 1963)

The present paper is a continuation of the work of A. G. Postnikov \((^1)\). In \((^1)\) the following was proved.

Theorem. Let \(g \geqslant 2\) be a natural number. Let \(p\) be a prime number; \(h=h(p)\) some integer-valued function; \(h \to \infty\) as \(p \to \infty\); \(h \leqslant \log p/2\log g\). Let \(\lambda>0\) be a constant. Denote by \(N_p(\lambda)\) the number of integers \(a\), \(0 \leqslant a \leqslant p-1\), for which

\[ \left|\sum_{x=0}^{h-1}\exp\left[2\pi i\,\frac{ag^x}{p}\right]\right|<\lambda\sqrt{h}. \]

Then, as \(p \to \infty\),

\[ \lim_{p\to\infty}\frac{1}{p}N_p(\lambda)=1-e^{-\lambda^2}. \]

In this theorem we have managed to remove the restriction \(h \leqslant \log p/2\log g\) in the case when the denominator has the special form \(p=g^\tau-1\), where \(\tau \geqslant 2\) is arbitrary natural. The result is formulated as the following theorem:

Theorem 1. Let \(g \geqslant 2\), \(\tau \geqslant 2\), \(c,h\) be natural numbers such that \(c\tau \to \infty\), \((c-1)\tau < h \leqslant c\tau\). Let \(\lambda>0\) be a constant. Denote by \(N_{g^\tau-1}(\lambda)\) the number of integers \(a\), \(0 \leqslant a \leqslant g^\tau-2\), for which

\[ \left|\sum_{x=0}^{h-1}\exp\left[2\pi i\,\frac{ag^x}{g^\tau-1}\right]\right| <\lambda\sqrt{(2c-1)h-c(c-1)\tau}. \]

Then, as \(c\tau \to \infty\),

\[ \lim_{c\tau\to\infty}\frac{1}{g^\tau-1}N_{g^\tau-1}(\lambda)=1-e^{-\lambda^2}. \]

The proof of this theorem is carried out by the method of moments in exactly the same way as was done in \((^1)\); however, the moments here are expressed in terms of the number of solutions of the congruence

\[ g^{x_1}+\ldots+g^{x_n}\equiv g^{y_1}+\ldots+g^{y_n}\pmod{g^\tau-1} \tag{1} \]

in integers \(x_i,y_i\) belonging to a longer interval than in \((^1)\). We find this number in Theorem 2; therefore, for the proof of Theorem 1 it is only necessary to repeat the reasoning of A. G. Postnikov in \((^1)\), but now taking into account the assertion of Theorem 2.

Theorem 2. Let \(n \geqslant 2\), \(g \geqslant 2\), \(\tau \geqslant 2\), \(c,h\) be natural numbers such that \(c\tau \to \infty\), \((c-1)\tau < h \leqslant c\tau\). Then for the number of solutions \(A_n(h)\) of congruence (1) in integers \(0 \leqslant x_i,y_i \leqslant h-1\) \((i,j=1,\ldots,n)\), as \(c\tau \to \infty\) the asymptotic formula

\[ A_n(h)=n!\,[(2c-1)h-c(c-1)\tau]^n+O(c^n h^{n-1}), \]

holds, where the constant in the \(O\)-symbol depends only on \(n\).

For the proof of this theorem we shall need the following

Lemma. Let \(g \geqslant 2\), \(n,c,p,N\) be natural numbers such that \(g<p\), \((g,p)=1\). Then the number of solutions of the congruence

\[ N\equiv g^{z_1}+\ldots+g^{z_n}\pmod p \tag{2} \]

in nonnegative integers \(z_1,\ldots,z_n\) such that \(g^{z_1}+\ldots+g^{z_n}\leqslant\)

\(\leq cp\), is estimated as \(O(c)\), where the constant in the symbol \(O\) depends only on \(n\).

Proof of the lemma. We divide all solutions of congruence (2) into \(c\) groups. To the \(k\)-th group \((k=1,2,\ldots,c)\) we assign those solutions \(z_1,\ldots,z_n\) for which

\[ (k-1)p<g^{z_1}+\cdots+g^{z_n}\leq kp. \tag{3} \]

Let \(z_1^{(k)},\ldots,z_n^{(k)}\) be some solution of congruence (2) belonging to the \(k\)-th group. Denote

\[ N_k=g^{z_1^{(k)}}+\cdots+g^{z_n^{(k)}}. \]

For any other solution \(\widetilde z_1^{(k)},\ldots,\widetilde z_n^{(k)}\) of congruence (2) belonging to the \(k\)-th group, we have

\[ N_k\equiv g^{\widetilde z_1^{(k)}}+\cdots+g^{\widetilde z_n^{(k)}}\pmod p. \tag{4} \]

In view of condition (3), congruence (4) is simply the equation

\[ N_k=g^{\widetilde z_1^{(k)}}+\cdots+g^{\widetilde z_n^{(k)}}. \tag{5} \]

In the work (²) A. G. Postnikov proved that the number of solutions of an equation of the form (5) in nonnegative integers is estimated as \(O(1)\), where the constant in the symbol \(O\) depends only on \(n\). Thus, in each of the \(c\) groups there will be \(O(1)\) solutions. Consequently, the number of solutions of congruence (2) is estimated as \(O(c)\), where the constant in the symbol \(O\) depends only on \(n\). The lemma is proved.

Proof of Theorem 2. Denote

\[ g^\tau-1=p. \tag{6} \]

We divide all solutions of congruence (1) into two categories:

1) Solutions \(x_1,\ldots,x_n,y_1,\ldots,y_n\) in which at least one of the relations \(x_i\equiv x_j\pmod \tau\) or \(y_i\equiv y_j\pmod \tau\) holds for \(i\ne j\). It is possible to specify no more than \(2\cdot C_n^2=O(1)\) different variants of such relations. Consider one of the variants, for example the case \(x_1\equiv x_2\pmod \tau\). In view of the condition \(0\leq x_1,x_2\leq h-1\leq c\tau-1\), for any value of \(x_2\) we can find no more than \(c\) values of \(x_1\) such that \(x_1\equiv x_2\pmod \tau\). Therefore, assigning to the quantities \(x_1,\ldots,x_n\) all possible values (from \(0\) to \(h-1\)), we obtain no more than \(ch^{n-1}\) systems \(x_1,\ldots,x_n\) for which \(x_1\equiv x_2\pmod \tau\). To each such system \(x_1,\ldots,x_n\), by the lemma there will correspond \(O(c)\) systems \(y_1,\ldots,y_n\). Thus, the number of solutions of congruence (1) belonging to the first category is estimated as \(O(c^2h^{n-1})\), where the constant in the symbol \(O\) depends only on \(n\).

2) Solutions \(x_1,\ldots,x_n,y_1,\ldots,y_n\) in which, for \(i\ne j\),

\[ x_i\not\equiv x_j\pmod \tau,\qquad y_i\not\equiv y_j\pmod \tau. \tag{7} \]

By \(\{a_1,a_2,a_3,\ldots\}\) we shall denote the set consisting of the numbers \(a_1,a_2,a_3,\ldots\). The range of variation of the quantities \(x_i\) and \(y_j\), i.e. the set \(M=\{0,1,\ldots,h-1\}\), is the union of the following \(c\) sets:

\[ M_1=\{0,1,\ldots,\tau-1\},\quad M_2=\{\tau,\ldots,2\tau-1\},\ldots,\quad M_{c-1}=\{(c-2)\tau,\ldots,(c-1)\tau-1\},\quad M_c=\{(c-1)\tau,\ldots,h-1\}. \]

Each of the first \(c-1\) sets \(M_i\) consists of \(\tau\) numbers, while the set \(M_c\) consists of \(t=h-(c-1)\tau\) numbers. Since \((c-1)\tau<h\leq c\tau\), we have \(1\leq t\leq \tau\).

Each of the first \(c-1\) sets \(M_i\) is divided into two subsets \(M_i^{(1)}\) and \(M_i^{(2)}\), assigning to \(M_i^{(1)}\) the first \(t\) numbers of the set \(M_i\), and to \(M_i^{(2)}\) the last \(\tau-t\) numbers of the set \(M_i\). Let \(M^{(1)}\) be the union of the sets \(M_1^{(1)},M_2^{(1)},\ldots,M_{c-1}^{(1)},M_c\), and \(M^{(2)}\) the union of the sets \(M_1^{(2)},M_2^{(2)},\ldots,M_{c-1}^{(2)}\). The set \(M^{(1)}\) will consist of \(ct\) numbers, and the set \(M^{(2)}\) of \((c-1)(\tau-t)\) numbers. The numbers belonging to the set \(M^{(1)}\) can be divided into \(t\) groups so that each group contains \(c\) numbers, and all the numbers,

belonging to one and the same group will be congruent to one another modulo \(\tau\). Similarly, the numbers belonging to the set \(M^{(2)}\) can be divided into \(\tau-t\) groups so that in each group there will be \(c-1\) numbers, and all numbers belonging to one and the same group will be congruent to one another modulo \(\tau\).

By \(\widetilde a\) we shall denote the least nonnegative residue of the integer \(a\) modulo \(\tau\). Take an arbitrary fixed system \(x_1,\ldots,x_n\) satisfying condition (7), and consider the number \(N=g^{x_1}+\cdots+g^{x_n}\). Since, by condition (6), \(g^\tau=p+1\), we have
\[ N \equiv g^{\widetilde{x}_1}+\cdots+g^{\widetilde{x}_n}\pmod p, \]
and \(\widetilde{x}_i\ne \widetilde{x}_j\) for \(i\ne j\). Further,
\[ 1\leq g^{\widetilde{x}_1}+\cdots+g^{\widetilde{x}_n}\leq g^{\tau-1}+g^{\tau-2}+\cdots+g^{\tau-n}<g^\tau=p+1, \]
i.e.
\[ 1\leq g^{\widetilde{x}_1}+\cdots+g^{\widetilde{x}_n}\leq p. \tag{8} \]

Let, among the numbers \(x_1,\ldots,x_n\) that we have taken, the numbers \(x_1,\ldots,x_m\in M^{(1)}\), and the numbers \(x_{m+1},\ldots,x_n\in M^{(2)}\), where \(0\leq m\leq n\) (if \(m=0\), then all the numbers are taken from \(M^{(2)}\), and if \(m=n\), then all the numbers are taken from \(M^{(1)}\)). Consider the congruence
\[ N\equiv g^{y_1}+\cdots+g^{y_n}\pmod p \tag{9} \]
in integers \(0\leq y_i\leq h-1\), satisfying condition (7). Congruence (9) is equivalent to the following:
\[ g^{\widetilde{x}_1}+\cdots+g^{\widetilde{x}_n} \equiv g^{y_1}+\cdots+g^{y_n}\pmod p. \tag{10} \]
By condition (8) and the fact that \(g^\tau=p+1\), the solutions of congruence (10), and consequently also of congruence (9), will be, up to a permutation of the elements, only those systems \(y_1,\ldots,y_n\) for which \(y_i\equiv x_i\pmod \tau\) \((i=1,\ldots,n)\). Since \(x_1,\ldots,x_m\in M^{(1)}\), \(x_{m+1},\ldots,x_n\in M^{(2)}\), each of the quantities \(y_1,\ldots,y_m\) can assume \(c\) values from \(M\) (more precisely, from \(M^{(1)}\)), and each of the quantities \(y_{m+1},\ldots,y_n\) can assume \(c-1\) values from \(M\) (more precisely, from \(M^{(2)}\)). Consequently, the number of solutions of congruence (9) is equal to \(n!c^m(c-1)^{n-m}\). Further, the number of distinct systems \(x_1,\ldots,x_n\) satisfying condition (7), and in which \(m\) numbers are taken from \(M^{(1)}\), and \(n-m\) numbers from \(M^{(2)}\), is equal to
\[ C_n^m(ct)(ct-1)\cdots(ct-m+1)[(c-1)(\tau-t)][(c-1)(\tau-t)-1]\cdots \]
\[ \cdots[(c-1)(\tau-t)-(n-m)+1] = C_n^m(ct)^m[(c-1)(\tau-t)]^{\,n-m} \]
\[ {}+O(c^{\,n-1}\tau^{\,n-1}) = C_n^m(ct)^m[(c-1)(\tau-t)]^{\,n-m}+O(h^{\,n-1}). \]
Therefore the number of solutions of congruence (1) belonging to the second category is equal to
\[ \sum_{m=0}^{n} n!c^m(c-1)^{n-m} \left\{C_n^m(ct)^m[(c-1)(\tau-t)]^{\,n-m}+O(h^{\,n-1})\right\} = \]
\[ = n![c^2t+(c-1)^2(\tau-t)]^n+O(c^nh^{\,n-1}) = n![(2c-1)h-c(c-1)\tau]^n+ \]
\[ {}+O(c^nh^{\,n-1}), \]
where the constant in the \(O\)-symbol depends only on \(n\).

Adding the numbers of solutions of congruence (1) belonging to both categories, we obtain the assertion of the theorem, since, evidently, as \(c\tau\to\infty\), in view of \(n\geq 2\) we have
\[ c^nh^{\,n-1}/[(2c-1)h-c(c-1)\tau]^n=O(1/c\tau). \]
The theorem is proved.

Omsk Machine-Building Institute

Received
16 I 1963

CITED LITERATURE

1. A. G. Postnikov, DAN, 133, 1289 (1960).
2. A. G. Postnikow, Festschrift anlässlich des 250 Geburtstages Leonhard Eulers, Berlin, 1959, S. 281.