R. Cohen proved the immersion conjecture in a 1985 Annals paper:

<cite authors="Cohen, Ralph L.">_Cohen, Ralph L._, [**The immersion conjecture for differentiable manifolds**](http://dx.doi.org/10.2307/1971304), Ann. Math. (2) 122, 237-328 (1985). [ZBL0592.57022](https://zbmath.org/?q=an:0592.57022).</cite>

_Any smooth compact n-dimensional manifold admits an immersion into Euclidean space of dimension 2n-a(n), where a(n) is the number of 1's in the binary decomposition of n._

Is there any result of this kind for vector bundles E over compact manifolds? Notice that the sphere bundle of E is compact. Maybe there is a silly argument...