A note on infinite antichain density

Let $\mathcal{F}$ be an antichain of finite subsets of $\mathbb{N}$. How quickly can the quantities $\mathrm{\mid }\mathcal{F}\cap 2^{[n]}\mathrm{\mid }$ grow as $n\to \mathrm{\infty }$? We show that for any sequence $(f_n{)}_{n\ge n_0}$ of positive integers satisfying ${\sum }_{n=n_0}^{\mathrm{\infty }}{f}_n\mathrm{/}{2}^n\le 1\mathrm{/}4$, $f_{n_0}=1$ and $f_n\le f_{n+1}\le 2f_n$,thereexistsaninfiniteantichain$$\mathcal{F}$$offinitesubsetsof$$\mathbb{N}$$suchthat$|$\mathcal{F}$\cap 2^{[n]}| \geq f_n$forall$n\ge n_0$. It follows that for any $\epsilon >0$ there exists an antichain $\mathcal{F}\subseteq 2^{\mathbb{N}}$ such that $$\underset{n\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}\mathrm{\mid }\mathcal{F}\cap 2^{[n]}\mathrm{\mid }\cdot {\left(\frac{2^n}{n{\log }^{1+\epsilon }n}\right)}^{-1}>0.$$ This resolves a problem of Sudakov, Tomon and Wagner in a strong form, and is essentially tight.