# A condition on $(a_{j})_{j\in \mathbb{N}}$ so that for all $x \in \mathbb{R}$ we have $\min_{1 \leq j \leq N}\|a_{j}x\|=o(1)$

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 $$j$$.

• I don't think so. You can fix an irrational $\theta$ and define the sequence greedily subject to $||a_d\theta|| > 1/3$, i.e., let $a_d$ be the smallest positive integer greater than $a_{d-1}$ that satisfies $d \mid a_d$ and $||a_d\theta|| > 1/3$. I think the resulting sequence will indeed have $\ge c\log N$ elements in $\{1,\dots,N\}$ for all (large) $N$. May 13, 2022 at 12:47
• Maybe a nice $\theta$ to consider would be an algebraic number like $\frac{1+\sqrt{5}}{2}$ and consider your approach. May 13, 2022 at 12:57
• Go ahead. Easy to code, for example. May 13, 2022 at 14:44

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).