Probabilistic results on the $2$-adic complexity

TL;DR

This paper investigates the average and asymptotic behavior of the 2-adic complexity of binary sequences, showing it is close to half the sequence length with high probability and providing bounds on its expected value.

Contribution

It establishes new bounds on the expected 2-adic complexity and proves almost sure asymptotic behavior for random binary sequences.

Findings

Expected 2-adic complexity is approximately N/2 with logarithmic deviation.

For random sequences, 2-adic complexity converges to N/2 with probability 1.

Provides bounds on the rational complexity of binary sequences.

Abstract

This work is devoted to solving some closely related open problems on the average and asymptotic behavior of the $2$-adic complexity of binary sequences. First, for fixed $N$, we prove that the expected value $E^{\mathrm{2-adic}}_N$ of the $2$-adic complexity over all binary sequences of length $N$ is close to $\frac{N}{2}$ and the deviation from $\frac{N}{2}$ is at most of order of magnitude $\log(N)$. More precisely, we show that $$\frac{N}{2}-1 \le E^{\mathrm{2-adic}}_N= \frac{N}{2}+O(\log(N)).$$ We also prove bounds on the expected value of the $N$th rational complexity. Our second contribution is to prove for a random binary sequence $\mathcal{S}$ that the $N$th $2$-adic complexity satisfies with probability $1$ $$ \lambda_{\mathcal{S}}(N)=\frac{N}{2}+O(\log(N)) \quad \mbox{for all $N$}. $$