Rational Approximation

Question

Suppose I have a very large integer $N$ and $a_1,....a_k$ are integers $(a_i,N)=1$ for any $1\le i\le k.$ Suppose also that $k$ is small compared to $N$ (as small as we wish). Does there exist an integer such that $(x,N)=1$ and for all $1\le i\le k$ $$||\frac{xa_i}{N}||\le \frac{1}{4},$$ where $|| \cdot||$ denotes the distance to the closest integer. Note, that Dirichlet's theorem in k-dimensions does not seem to imply the result directly since we have an extra condition on $x$ being coprime to $N.$

Answer 1

Let $N=8$, $k=2$, $a_1=1$, $a_2=5$.

Then $a_1x/N=x/8$ and its distance from the nearest integer is

1. $1/8$ if $x$ is $8r\pm1$,

2. $3/8\gt1/4$ if $x$ is $8r\pm3$;

$a_2x/N=5x/8$ and its distance from the nearest integer is

1. $1/8$ if $x$ is $8r\pm3$,

2. $3/8\gt1/4$ if $x$ is $8r\pm1$.

So there is no $x$, $\gcd(x,N)=1$, such that both $a_1x/N$ and $a_2x/N$ are within $1/4$ of an integer.

A similar construction works for $N=12,16,20,\dots$.