MathDB

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.

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