MATHEMATICS

A. G. POSTNIKOV

ON A VERY SHORT EXPONENTIAL RATIONAL TRIGONOMETRIC SUM

(Presented by Academician I. M. Vinogradov on 15 IV 1960)

The present paper is an analogue of the work of M. P. Mineev (¹) and is a new application of A. A. Markov’s method of moments* to number-theoretic problems.

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

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

Theorem. As \(p\to\infty\),

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

Proof. For fixed \(p\) and, consequently, \(h\), consider the random variable \(\xi_p\), taking the values

\[ \frac{1}{h}\left|\sum_{x=0}^{h}\exp\left[2\pi i\,\frac{ag^x}{p}\right]\right|^2,\qquad a=0,1,\ldots,p-1, \]

with probability equal to \(1/p\). The distribution function of this random variable is \(N_p(\lambda^2)/p\).

Let us compute the \(r\)-th moment of this distribution function:

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

\[ =\frac{1}{ph^r} \sum_{x_1=0}^{h}\cdots\sum_{x_r=0}^{h} \sum_{y_1=0}^{h}\cdots\sum_{y_r=0}^{h} \exp\left[ 2\pi i\,\frac{a\left(g^{x_1}+\cdots+g^{x_r}-g^{y_1}-\cdots-g^{y_r}\right)}{p} \right] =\frac{1}{h^r}M_r(p), \]

where \(M_r(p)\) is the number of solutions of the congruence

\[ g^{x_1}+\cdots+g^{x_r}\equiv g^{y_1}+\cdots+g^{y_r}\pmod p \tag{1} \]

in the integers \(0\leqslant x_i,y_i\leqslant h\), \(i=1,2,\ldots,r\). But since \(0\leqslant x_i,y_i\leqslant h\leqslant \dfrac{1}{2}\dfrac{\log p}{\log g}\), we have \(r\leqslant g^{x_1}+\cdots+g^{x_r}\leqslant r\sqrt p\).

* An application of the method of moments similar in idea is found in the works (², ³).

If \(p>r^2\), then

\[ 0<g^{x_1}+\ldots+g^{x_r}<p, \]

and the congruence of the two sides of (1) means equality. Therefore \(M_r(p)\) is equal to the number of solutions of the equation

\[ g^{x_1}+\ldots+g^{x_r}=g^{y_1}+\ldots+g^{y_r} \tag{2} \]

in numbers \(0\leq x_i,y_i\leq h\).

Let \(r\) be fixed, \(p\to\infty\) (and hence \(h\to\infty\)). The number of solutions of equation (2) is expressed by the formula

\[ M_r(p)=r!\,h^r+O\left(h^{r-1}\right) \]

(a more general assertion was proved in paper \({}^{4}\)).

Thus,

\[ \lim_{p\to\infty}\frac1p\sum_{a=0}^{p-1}\frac1{h^r} \left|\sum_{x=0}^{h}\exp\left[2\pi i\frac{ag^x}{p}\right]\right|^{2r}=r! \]

Applying the second limit theorem of probability theory in the way this was done in paper \({}^{1}\), we obtain

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

which is what was required to prove.

Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR

Received
14 IV 1960

REFERENCES

\({}^{1}\) M. P. Mineev, UMN, 14, no. 3, 169 (1959).
\({}^{2}\) H. Davenport, P. Erdős, Publ. Math., 2, 252 (1952).
\({}^{3}\) I. P. Kubilius, Yu. V. Linnik, Izv. Higher Educational Institutions, Mathematics, no. 6, 88 (1959).
\({}^{4}\) M. P. Mineev, Matem. sbornik, 46 (88), 4, 451 (1958).