On the defining equations of the tangent cone of a numerical semigroup ring

Let $\mathbf{a} = a_1 <\dots < a_r$ be a sequence of positive integers, and let $H_{\mathbf{a}}$ denote the semigroup generated by $a_1, \dots, a_r$. For an integer $k\geq 0$ we denote by $\mathbf{a}+k$ the shifted sequence $a_1 +k, \dots, a_r +k$. Fix a field $K$. We show that for all $k \gg 0$ the associated graded ring of the semigroup ring $K[H_{\mathbf{a}+k}]$ is Cohen--Macaulay and that it has the same Betti numbers as $K[H_{\mathbf{a}+k}]$ itself. As a consequence, we show that the number of defining equations of the tangent cone of a numerical semigroup ring is bounded by a value depending only on the width of the semigroup, where the width of a numerical semigroup is defined to be the difference of the largest and the smallest element in the minimal generating set of the semigroup. We also provide a conjectured upper bound of the above number of equations and we verify it in some cases.