Question 167
Given a prime number $p$ and a positive real number $\lambda$ less than $1$. Let $k$ be an integer, and let $S$ and $T$ be sets consisting of consecutive $s$ and $t$ integers, respectively, satisfying $1 \le s \le t \lt \frac{\lambda}{12}p$. Furthermore, assume that the number of elements in the set
$$
\{(x, y) \in S \times T : kx \equiv y \pmod{p}\}
$$
is not less than $1 + \lambda s$. Prove that there exist integers $a$ and $b$ satisfying
$$
ka \equiv b \pmod{p}, \quad 0 \lt a \le \frac{1}{\lambda}, \quad |b| \le \frac{t}{\lambda s}.
$$
$$
\{(x, y) \in S \times T : kx \equiv y \pmod{p}\}
$$
is not less than $1 + \lambda s$. Prove that there exist integers $a$ and $b$ satisfying
$$
ka \equiv b \pmod{p}, \quad 0 \lt a \le \frac{1}{\lambda}, \quad |b| \le \frac{t}{\lambda s}.
$$