SCHWEITZER 2021 Problem 7

If the binary representations of the positive integers $k$ and $n$ are $k = \sum_{i=0}^{\infty} k_i 2^i$ and $n = \sum_{i=0}^{\infty} n_i 2^i$ , then the logical sum of these numbers is $$k \oplus n =\sum_{i=0}^{\infty} |k_i-n_i|2^i.$$ Let $N$ be an arbitrary positive integer and $(c_k)_{k \in \mathbb{N}}$ be a sequence of complex numbers such that for all $k \in \mathbb{N}$ , $|c_k| \le 1$ . Prove that there exist positive constants $C$ and $\delta$ such that $$\int_{[-\pi,\pi] \times [-\pi, \pi]} \sup_{n<N, n \in \mathbb{N}} \frac{1}{N} \Big| \sum_{k=1}^{n} c_k e^{i(kx+(k \oplus n) y)} \Big| \mathrm d(x,y) \le C \cdot N^{-\delta}$$ holds.