Lempel-Ziv: a "one-bit catastrophe" but not a tragedy

The so-called "one-bit catastrophe" for the compression algorithm LZ'78 asks whether the compression ratio of an infinite word can change when a single bit is added in front of it. We answer positively this open question raised by Lutz and others: we show that there exists an infinite word $w$ such that $\rho_{sup}(w)=0$ but $\rho_{inf}(0w)>0$, where $\rho_{sup}$ and $\rho_{inf}$ are respectively the $\limsup$ and the $\liminf$ of the compression ratios $\rho$ of the prefixes. To that purpose we explore the behaviour of LZ'78 on finite words and show the following results: - There is a constant $C>0$ such that, for any finite word $w$ and any letter $a$, $\rho(aw)\leq C\sqrt{\rho(w)\log|w|}$. Thus, sufficiently compressible words ($\rho(w)=o(1/\log|w|)$) remain compressible with a letter in front; - The previous result is tight up to a multiplicative constant for any compression ratio $\rho(w)=O(1/\log|w|)$. In particular, there are infinitely many words $w$ satisfying $\rho(w)=O(1/\log|w|)$ but $\rho(0w)=\Omega(1)$.