Recall the well-known result of Whitney that any (compact) smooth \(n\)-manifold admits an immersion into \(\mathbb{R}^{2n-1}\). Today there was a preprint posted on the arXiv (arXiv:2011.00974) which mentioned in its introduction the following result of Cohen, which strengthens Whitney’s result as follows:

Any (compact) smooth \(n\)-manifold admits an immersion into \(\mathbb{R}^{2n-\alpha(n)}\), where \(\alpha(n)\) is the number of ones in the dyadic expression of \(n\).

That is to say, writing \(n = 2^{i_1} + \cdots + 2^{i_l}\) with \(i_1 < \ldots < i_l\), then \(\alpha(n) = l\).

Cohen’s paper was published in 1985, but up to now I was not aware of his result …

It even has an article in German Wikipedia: https://de.wikipedia.org/wiki/Immersionssatz_von_Cohen