MathDB

Given a prime

p

and a real number

\lambda \in (0,1)

. Let

s

and

t

be positive integers such that

s \leqslant t < \frac{\lambda p}{12}

S

and

T

are sets of

s

and

t

consecutive positive integers respectively, which satisfy

\left| \left\{ (x,y) \in S \times T : kx \equiv y \pmod p \right\}\right| \geqslant 1 + \lambda s.

Prove that there exists integers

a

and

b

that

1 \leqslant a \leqslant \frac{1}{ \lambda}

\left| b \right| \leqslant \frac{t}{\lambda s}

and

ka \equiv b \pmod p

P23