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).

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 $j$.