Erdős Problem #26 — A shift $k$ making the multiples of $n_i+k$ have density 1

Erdős Problem #26 asks whether every infinite set $A$ admits a shift $A+k$ whose members divide almost all integers. The answer is negative; a simple counterexample follows from classical work, with stronger counterexamples also recorded.

Known results

• Davenport–Erdős (1951), with the result also associated with Tenenbaum: every Behrend sequence has divergent reciprocal sum, so any infinite $A$ with $\sum_{a\in A}1/a<\infty$ disproves the question.
• Ruzsa gave a counterexample in which no translate $A+k$ is a Behrend sequence.
• Van Doorn explained how to modify Ruzsa’s construction to obtain a counterexample with $\sum_{a\in A}1/a=\infty$.

Reported DeepMind result for Tenenbaum’s variant

A DeepMind prover agent reportedly constructed an infinite $A$ such that, for every $k\geq1$, the multiples of $A+k$ have upper density $<0.34$, disproving the weaker $1-\epsilon$ formulation if verified. The associated Lean theorem is displayed with $\texttt{sorry}$.

Current status (as of June 2026): the original problem is resolved negatively; Tenenbaum’s weaker density variant is claimed disproved by DeepMind but remains unverified.