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Glasner sets and uniform dilations

In [1], S. Glasner proved the following result: Let $A$ be an infinite subset of $\mathbb T=\mathbb R/\mathbb Z$ . Then for every $\epsilon>0$ , there exists an integer $n$ such that $nA:=\{na: a \in A\}$ intersects every interval of length $\epsilon$ in $\mathbb T$ (in this case, $nA$ is said to be $\epsilon$ -dense in $\mathbb T$ ).
In [2], a set of integers $S$ was defined to be a Glasner set if for every infinite $A \subset \mathbb T$ and $\epsilon>0$ , there exists $n \in S$ such that $nA$ is $\epsilon$ -dense in $\mathbb T$ . It was shown in [2] that if $P$ is a nonconstant polynomial with integer coefficients, then $\{P(n): n \in \mathbb Z\}$ is a Glasner set, yet a finite union of lacunary sequences is never a Glasner set. In [3] it was proved that the primes form a Glasner set, and this was extended in [4] to polynomial images of the primes.

Question 1: Are there two sets of integers neither of which is a Glasner set, but whose union does form a Glasner set?

A quantitative version of Glasner's result was established in [2], [3]: Given $\epsilon>0$ , there exists an integer $k$ such that, for any set $A \subset \mathbb T$ of cardinality at least $k$ , there is an integer $n$ such that $nA$ is $\epsilon$ -dense in $\mathbb T$ . If the minimal such $k$ is denoted by $k(\epsilon)$ , then for some absolute constant $c$ ,

$\frac{c}{\epsilon^2} \leq k(\epsilon) \leq \epsilon^{-2+o(1)}$

as $\epsilon \to 0$ . The lower bound is obtained by taking $A=A(\epsilon)$ consisting of all reduced fraction $j/\ell$ where $1 \le j \le \ell < \epsilon^{-1}$ .
The upper bound is harder to prove; but could the lower bound be sharp?

Question 2: Is $k(\epsilon) =O(\epsilon^{-2})$ ?

[1] Glasner, Shmuel.
Almost periodic sets and measures on the torus.
Israel J. Math. 32 (1979), no. 2-3, 161–-172.

[2] Berend, Daniel and Peres, Yuval.
Asymptotically dense dilations of sets on the circle.
J. London Math. Soc. (2) 47 (1993), no. 1, 1–-17.

[3] N. Alon and Y. Peres, "Uniform dilations'', Geometric and Functional Analysis, vol. 2, No. 1 (1992), 1–28.

[4] Nair, R. and Velani, S. L.
Glasner sets and polynomials in primes.
Proc. Amer. Math. Soc. 126 (1998), no. 10, 2835–-2840.

Glasner sets and uniform dilations

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