Some properties of a Rudin-Shapiro-like sequence

TL;DR

This paper introduces a binary sequence based on inversions, analyzes its properties, and demonstrates that its partial sums exhibit oscillatory behavior proportional to the square root of N, similar to the Rudin-Shapiro sequence.

Contribution

The paper defines a new sequence related to binary inversions and characterizes its partial sum behavior, revealing oscillations akin to the Rudin-Shapiro sequence.

Findings

Partial sums grow proportionally to √N with oscillations

Sequence exhibits properties similar to Rudin-Shapiro sequence

Oscillation bounds between √3/3 and √2

Abstract

We introduce the sequence $(i_n)_{n \geq 0}$ defined by $i_n = (-1)^{inv_2(n)}$, where $inv_2(n)$ denotes the number of inversions (i.e., occurrences of 10 as a scattered subsequence) in the binary representation of n. We show that this sequence has many similarities to the classical Rudin-Shapiro sequence. In particular, if S(N) denotes the N-th partial sum of the sequence $(i_n)_{n \geq 0}$, we show that $S(N) = G(\log_4 N)\sqrt{N}$, where G is a certain function that oscillates periodically between $\sqrt{3}/3$ and $\sqrt{2}$.