Let $\boldsymbol{A}=[\boldsymbol{a_1},\boldsymbol{a_2},\ldots,\boldsymbol{a_T}]^\text{tr}\in\mathbb{R}^{T\times D}$, and $S$ be the integer lattice points within and on the surface of a convex body. In particular, the shape of interest to me for $S$ is $T$-dimensional $L^1$-disc, that is $S=\{(s_1,\ldots,s_T)\in\mathbb{Z}^T|\sum_{i=1}^{T}|s_i|\leq R\}$.
I want to analyze the cardinality of $A(S)$ (image of $S$ under $A$). $$\#A(S)=\#\Big\{\sum_{i=1}^{T}s_i\boldsymbol{a_i}\Big|(s_1,\ldots,s_T)\in S\Big\}$$ If $\boldsymbol{a_i}$'s are linearly independent, then $\#A(S)=\#S$. However, if $T>D$, or $det(A)=0$, this method doesn't work, as there may be points in $S$ with the same image under $A$. I'm wondering if there are any theorems or techniques from additive combinatorics or the geometry of numbers to solve this problem.