Lattices with congruence densities larger than $3\mathrm{/}32$

By a 1997 result of R. Freese, an $n$-element lattice has at most $2^{n-1}$ congruences. This motivates us to define the congruence density cd$(L)$ of a finite $n$-element lattice as $\mathrm{\mid }$Con$(L)\mathrm{\mid }\mathrm{/}{2}^{n-1}$, where $\mathrm{\mid }$Con$(L)\mathrm{\mid }$ is the number of elements of the congruence lattice Con$(L)$ of $L$. We prove that whenever $L$ is a finite lattice with cd$(L)>3\mathrm{/}32$, then $L$ has the same number of join-irreducible and meet-irreducible elements. This result is sharp, since there exists a six-element lattice $R_6$ with cd$(R_6)=3\mathrm{/}32$ but fewer join-irreducible than meet-irreducible elements. By R. Freese, C. Mureşan, J. Kulin, and the present author’s results, lattices with congruence densities larger than $1\mathrm{/}8$ have already been described. Here we decrease the lower threshold from $1\mathrm{/}8$ to $3\mathrm{/}32$. That is, we describe all finite lattices $L$ such that cd$(L)>3\mathrm{/}32$. As a corollary, we give the $k$th largest number of congruences of $n$-element lattices for $n>8$ and $k\in \{n+1,n+2,n+3,n+4\}$.