Let $A, B, C \subseteq \mathbb{Z}_2^n$. Define $r(A, B, C) := |\{(a, b, c) \in A \times B \times C : a + b = c\}|$.
For $1 \leq a \leq 2^n - 1$, let $A_a \subseteq \mathbb{Z}_2^n$ denote the set corresponding to the binary representations of the numbers $\{2^n - a, \dots, 2^n - 1\}$.
For example, if $n = 3$ and $a = 5$, then $A_5 = \{011, 100, 101, 110, 111\}$ and $A_7=\{001,010,011,100,101,110,111\}$ for $a=7$.
Now, let $1 \leq s \leq 2^{n-1}$ and $2^{n-1} < t \leq 2^n - 1$. I am wondering whether it is possible to compute the value of $r(A_s, A_s, A_t)$ directly.
For example, it is trivial to see that if $1 \leq s, t \leq 2^{n-1}$, then $r(A_s, A_s, A_t) = 0.$ Similarly, it is not difficult to show that if $2^{n-1} < s \leq 2^n - 1$ and $1 \leq t \leq 2^{n-1}$, then $$r(A_s, A_s, A_t) = 2st - 2^n t.$$
