Tangent cones of monomial curves obtained by numerical duplication

Given a numerical semigroup ring $R=k[\![S]\!]$, an ideal $E$ of $S$ and an odd element $b \in S$, the numerical duplication $S \! \Join^b \! E$ is a numerical semigroup, whose associated ring $k[\![S \! \Join^b \! E]\!]$ shares many properties with the Nagata's idealization and the amalgamated duplication of $R$ along the monomial ideal $I=(t^e \mid e\in E)$. In this paper we study the associated graded ring of the numerical duplication characterizing when it is Cohen-Macaulay, Gorenstein or complete intersection. We also study when it is a homogeneous numerical semigroup, a property that is related to the fact that a ring has the same Betti numbers of its associated graded ring. On the way we also characterize when ${\rm gr}_{\mathfrak m}(I)$ is Cohen-Macaulay and when ${\rm gr}_{\mathfrak m}(\omega_R)$ is a canonical module of ${\rm gr}_{\mathfrak m}(R)$ in terms of numerical semigroup's properties, where $\omega_R$ is a canonical module of $R$.