Suppose that the sequence $(a\_{j})\_{j \in \mathbb{N}}$ is an increasing sequence of positive integers that satisfies $$(1)\text{ }\text{ }\text{ }\text{ }\text{ }\text{ } d | a\_{d}$$ and $$ (2)\text{ }\text{ }\text{ }\text{ } |\{v| v\in (a\_{j})\_{j \in \mathbb{N}}, v \leq N \}| \geq C\log(N).$$
for a given constant $C > 0$ and every $N \in \mathbb{N}$. For a given $x \in \mathbb{R}$, do we have


$$(3)\text{ }\text{ }\text{ }\text{ }\text{ } \lim\_{N \rightarrow \infty}\min\_{1 \leq j \leq N}\|a\_{j}x\| = 0$$


(Here $\|y\|$ refers to the distance from $y$ to the nearest integer).




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Note that if the sequence $(a\_{j})\_{j \in \mathbb{N}}$ satisfies the property $(3)$ then it is a Heilbronn sequence. <https://en.wikipedia.org/wiki/Heilbronn_set>


A Heilbronn sequence must contain multiples of every number, furthermore, this is not a sufficient condition. If the underlying sequence is sparse like $(j!)\_{j \in \mathbb{N}}$ then it is not a Heilbronn sequence.


This conjecture is related to Dirichlet's approximation theorem, the paper "Small values of $n^2 \alpha$ $\mod 1$" and "Analytischer Beweis des Satzes uber die Verteilung der Bruchteile eines ganzen Polynoms" where they consider the case where $a\_{j}$ is some fixed power of $j$.



A condition on $(a_{j})_{j\in \mathbb{N}}$ so that for all $x \in \mathbb{R}$ we have $\min_{1 \leq j \leq N}\|a_{j}x\|=o(1)$
I will expand upon @mathworker21's comment.


Let $\theta = \frac{1 + \sqrt{5}}{2}$, and define the sequence $a\_d$ greedily to be the smallest positive integer greater than $a\_{d - 1}$ such that $d | a\_d$ and $\|a\_d \theta\| > \frac{1}{3}$.


Let $m$ be the smallest multiple of $d$ which is greater than $a\_{d - 1}$. Clearly, $m \leq a\_{d - 1} + d$. To find $a\_d$, we take the number $m \theta$, and repeatedly add $d \theta$ until we get a number whose fractional part is between $\frac{1}{3}$ and $\frac{2}{3}$.


By Dirichlet's approximation theorem, for some $1 \leq k \leq 3$ we have
$$\| kd \theta \| \leq \frac{1}{3}$$
and it is well known (for this specific value of $\theta$) that we have
$$\| kd \theta \| \geq \frac{1}{3 k^2 d^2} \geq \frac{1}{27 d^2}.$$
Therefore, if we take $m \theta$ and add $kd \theta$ repeatedly, after at most $27 d^2$ steps we will get a number with fractional part between $\frac{1}{3}$ and $\frac{2}{3}$, which shows that
$$a\_d \leq a\_{d - 1} + 30 d^3.$$
By induction, we have
$$a\_d \leq 30 d^4$$
which is sufficient for condition (2).


