We show that the only degree which cannot co-enumerate a non-trivial $\Pi^0_1$-immune real is $\emptyset$, resolving a conjecture of the author. The main ingredient is that if a real $A$ has that all $A$-maximal sets are c.e., then $A$ has c.e. degree, which can be proved via a coding mechanism and two classical results of Lachlan.
Let $G$ be a finite group and $x \in G$. In 2013, Hai-Reuven proved that the cardinality of the centralizer of $x$ divides the cardinality of the solubilizer of $x$. Settling a conjecture of Mousavi, Poozesh and Zamani, we show that the cardinality of the normalizer of $\langle x \rangle$ is also a divisor. We actually...
We show that there exists a constant $\epsilon>0$ such that if $C$ is a conjugacy class in $A_n$ of size at least $|A_n|^{1-\epsilon}$, then $A_n \setminus \{1\} \subseteq C^2$. This proves the robust version of Thompson's conjecture for the alternating group, conjectured by Shalev (Annals of Math., 2009). Our proof co...
Nathan Keller, Noam Lifshitz, Avichai Marmor et al.· 1 citation
Let $q_n$ denote the $n$th product of two distinct primes. By completing a square in Sono's sieve and solving the resulting one-dimensional variational problem, we prove unconditionally that, for every $\varepsilon>0$ and all sufficiently large integers $\rho$, \[ \liminf_{n\to\infty}(q_{n+\rho}-q_n) \leq \exp\left((\p...
The Kannan--Lov\'asz--Simonovits (KLS) conjecture asks whether every isotropic log-concave probability measure on $\mathbb R^n$ has a Cheeger constant bounded below by a universal positive constant. We prove that $\psi_n\le C$ for a universal constant $C>0$, resolving the KLS conjecture. We also prove that $C_P(\mu)\le...
The least primitive root $g(p)$ modulo a prime $p$ is conjectured by Grosswald to satisfy $g(p)<\sqrt{p}-2$ for any $p>409$. We make progress towards the conjecture by proving $g(p)<\sqrt{p}-2$ when $p>1.77 \times 10^{54}$.
A subsemiring $S$ of the nonnegative cone of the real line containing $1$ is called a positive semiring. The Bi-UF Positive Conjecture, posed by Baeth, Chapman, and Gotti in 2021, states that the prototypical semiring $\mathbb{N}_0$ is the only positive semiring whose additive monoid $(S,+)$ and nonzero multiplicative...
Omar Graia· 0 citations
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