Erdős Problem #1123

The problem

Let $B_1$ be the Boolean algebra of sets of integers modulo sets of density $0$ (that is, in which two sets are equivalent if and only if they differ by a set of density $0$) and let $B_2$ be the Boolean algebra of sets modulo sets of logarithmic density $0$.

Prove that $B_1$ and $B_2$ are not isomorphic.

Statement quoted from erdosproblems.com/1123, maintained by Thomas Bloom. Original source: [VMR80,p.238][Er81b,p.30]. Current status there: independent.

AI contributions

As recorded in Terence Tao's wiki AI contributions to Erdős problems (data as of 30 Jun 2026). 🟢 full, 🟡 partial, ⚪ unverified, 🔴 incorrect.

Caveats

The wiki's classifications are provisional and are not a benchmark. Many Erdős problems have never had a thorough literature review, so a result that looks new may later turn out to be known; some problems were also stated ambiguously or incorrectly. See the problem page for the full discussion.

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