Oleg Pikhurko

Oleg Pikhurko, Oleg Verbitsky, Maksim Zhukovskii

We prove that the density of any covering single-insertion code $C\subseteq X^r$ over the $n$-symbol alphabet $X$ cannot be smaller than $1/r+δ_r$ for some positive real $δ_r$ not depending on $n$. This improves the volume lower bound of $1/(r+1)$. On the other hand, we observe that, for all sufficiently large $r$, if $n$ tends to infinity then the asymptotic upper bound of $7/(r+1)$ due to Lenz et al (2021) can be improved to $4.911/(r+1)$. Both the lower and the upper bounds are achieved by relating the code density to the Turán density from extremal combinatorics. For the last task, we use the analytic framework of measurable subsets of the real cube $[0,1]^r$.

Endre Csóka, Łukasz Grabowski, András Máthé et al.

We study a variant of the parallel Moser-Tardos Algorithm. We prove that if we restrict attention to a class of problems whose dependency graphs have subexponential growth, then the expected total number of random bits used by the algorithm is constant; in particular, it is independent from the number of variables. This is achieved by using the same random bits to resample variables which are far enough in the dependency graph. There are two corollaries. First, we obtain a deterministic algorithm for finding a satisfying assignment, which for any class of problems as in the previous paragraph runs in time O(n), where n is the number of variables. Second, we present a Borel version of the Lovász Local Lemma.