MathDB

[*]Prove that for every real number

t

such that

0 < t < \tfrac{1}{2}

there exists a positive integer

n

with the following property: for every set

S

n

positive integers there exist two different elements

x

and

y

S

, and a non-negative integer

m

(i.e.

m \ge 0

), such that

|x-my|\leq ty.

[*]Determine whether for every real number

t

such that

0 < t < \tfrac{1}{2}

there exists an infinite set

S

of positive integers such that

|x-my| > ty

for every pair of different elements

x

and

y

S

and every positive integer

m

(i.e.

m > 0

6