What does the function $f(x,y)$ reduce to? $$f(x,y) = \lim_{n \to \infty} \sum_{i = 0}^{n} (((x \bmod 2^{i-1})-(x \bmod 2^i)) \cdot ((y \bmod 2^{i-1})-(y \bmod 2^i)) \cdot \frac {1}{2^i})$$

The problem is proposed here and related to this question. Given $n$ and $k$, I would like to know how to compute$$\sum_{\substack{x_0 ⊕x_1⊕\cdots⊕x_k=0\\x_i≥0,\ 0≤i≤k\\\sum\limits_{i=0}^kx_i≤n-2k}}\binom{n-k-\sum\limits_{i=0}^kx_i}k$$ in $O(nk·\log n)$ time, where $⊕$ is exclusive or. Let $\sum...

Let $Z,Z'\in\{-1,1\}^\ell$ be two independent vectors of i.i.d. Rademacher r.v.'s, where $1\leq \ell \leq d$ are two integers ($d\gg 1$). I am trying to get an upper bound on $$ \mathbb{E}_{ZZ'}[ e^{n\left(e^{\sum_{i=1}^\ell \mathrm{Erf}(\varepsilon Z_i /\sqrt{d})} -1\right)\left(e^{\sum_{i=1}^\e...