Consider two d-uplets $u = (u_1,...,u_d)$ and $v = (v_1, ..., v_d)$ both living in $\mathbb{N}^d$ with $d$ a positive integer. They both verify $$(*) \sum_{i=1}^d u_i = \sum_{i=1}^d v_j = k$$ with $k$ a positive integer, $k\geq d$ supposed to tend to infinity later on. Given a constant $c \in \mathbb{N}$ such that $c<k$ ($c$ could also depend on $k$, e.g. $ c= \lceil k/d \rceil$), I would like to upper bound the cardinal of the following sets: $$\{(u,v) | u \neq v, (*), \sum_{i=1}^d |u_i -v_j| \leq c \}. $$

It could also be written as bounding the cardinal of the set $$\{ u | u \neq v, (*), \sum_{i=1}^d |u_i -v_j| \leq c \} $$ for any $v$.

My knowledge of combinatorics is very low, I'd be glad to receive any reference or idea that could lead to the solution of this problem or similar ones. I do not even have intuition for the magnitude order of this quantity.

Consider two d-uplets $u = (u_1,...,u_d)$ and $v = (v_1, ..., v_d)$ both living in $\mathbb{N}^d$ with $d$ a positive integer. They both verify $$(*) \sum_{i=1}^d u_i = \sum_{i=1}^d v_i = k$$ with $k$ a positive integer, $k\geq d$ supposed to tend to infinity later on. Given a constant $c \in \mathbb{N}$ such that $c<k$ ($c$ could also depend on $k$, e.g. $ c= \lceil k/d \rceil$), I would like to upper bound the cardinal of the following sets: $$\{(u,v) | u \neq v, (*), \sum_{i=1}^d |u_i -v_i| \leq c \}. $$

It could also be written as bounding the cardinal of the set $$\{ u | u \neq v, (*), \sum_{i=1}^d |u_i -v_i| \leq c \} $$ for any $v$.

My knowledge of combinatorics is very low, I'd be glad to receive any reference or idea that could lead to the solution of this problem or similar ones. I do not even have intuition for the magnitude order of this quantity.