MathDB

Sets

X\subset \mathbb{Z}^{+}

and

Y\subset \mathbb{Z}^{+}

are called comradely, if every positive integer

n

can be written as

n=xy

for some

x\in X

and

y\in Y

. Let

X(n)

and

Y(n)

denote the number of elements of

X

and

Y

, respectively, among the first

n

positive integers.
Let

f\colon \mathbb{Z}^{+}\to \mathbb{R}^{+}

be an arbitrary function that goes to infinity. Prove that one can find comradely sets

X

and

Y

such that

\dfrac{X(n)}{n}

and

\dfrac{Y(n)}{n}

goes to

0

, and for all

\varepsilon>0

exists

n \in \mathbb{Z}^+

such that

\frac{\min\big\{X(n), Y(n)\big\}}{f(n)}<\varepsilon.

A.838