Birch's theorem with shifts

Let $f_1,\mathrm{.}\mathrm{.}\mathrm{.},f_R$ be rational forms of degree $d\ge 2$ in $n > \sigma + R(R+1)(d-1)2^{d-1}$variables,where$\sigma$ is the dimension of the affine variety cut out by the condition $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}(\mathrm{\nabla }{f}_k{)}_{k=1}^R<R$. Assume that $\mathbf{f}=\mathbf{0}$ has a nonsingular real solution, and that the forms $(1,\mathrm{.}\mathrm{.}\mathrm{.},1)\cdot \mathrm{\nabla }{f}_k$ are linearly independent. Let $\mathit{\tau }\in {\mathbb{R}}^R$, let $\mu$ be an irrational real number, and let $\eta$ be a positive real number. We consider the values taken by $\mathbf{f}(x_1+\mu ,\mathrm{.}\mathrm{.}\mathrm{.},x_n+\mu )$ for integers $x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n$. We show that these values are dense in ${\mathbb{R}}^R$, and prove an asymptotic formula for the number of integer solutions $\mathbf{x}\in [-P,P{]}^n$ to the system of inequalities $\mathrm{\mid }{f}_k(x_1+\mu ,\mathrm{.}\mathrm{.}\mathrm{.},x_n+\mu )-{\tau }_k\mathrm{\mid }<\eta$ ($1 \le k\le R$).