An improvement of the Hash–Tognoli theorem

Let $\mathcal M$ be a smooth closed manifold embedded in $\mathbb R^n$. The Hash–Tognoli theorem asserts that if $\dim\mathcal M<(n-1)/2$ then $\mathcal M$ can be arbitrary well approximated (in the $C^r$-topology with $r<\infty$) in $\mathbb R^n$ by a nonsingular real algebraic set. There is a well-known conjecture going back to Hash which asserts that the restriction on $\dim\mathcal M$ in the Hash-Tognoli theorem is in fact superfluous. But so far the possibility of approximation in the nonstable dimensions (i. e. for $\dim\mathcal M\geqslant(n-1)/2$) was known only for orientable $\mathcal M$ with codimension (in $\mathbb R^n$) 1 and 2. The purpose of the paper is to prove the following theorem, which weakens the restriction on $\dim\mathcal M$ in the Hash–Tognoli theorem to $\dim\mathcal M<(2n-1)/3$.
Theorem. If $\mathcal M$ is a smooth closed manifold embedded in $\mathbb R^n$, and $\dim\mathcal M<(2n-1)/3$ then $\mathcal M$ can be arbitrary well approximated in $\mathbb R^n$ by a nonsingular real algebraic set.