Erdős Problem #1134

Let \(A\subseteq \mathbb{N}\) be the smallest set which contains \(1\) and is closed under the operations \[x\mapsto 2x+1,\quad x\mapsto 3x+1,\quad x\mapsto 6x+1.\] Does \(A\) have positive lower density? Lagarias [La16] reports that Erdős asked this in 1972, offering £10 for a solution. (Although Hilton told Lagarias that this problem may have been formulated by Klarner, and that Erdős liked it and offered a prize for its solution.) Erdős had earlier proved (as reported in [KlRa74]) that if \(A\) is the smallest set which contains \(1\) and is closed under the operations \(x\mapsto m_ix+b_i\) for some (possibly infinite) collection of \(m_i\geq 1\) and \(b_i\geq 0\) then, if \(\sigma>0\) is such that \(\sum \frac{1}{m_i^\sigma}=1\) then, for all large \(X\), \(\lvert A\cap [1,X]\rvert \ll X^{\sigma+o(1)}\). This result does not help with the given problem since \(\frac{1}{2}+\frac{1}{3}+\frac{1}{6}=1\). This was answered in the negative soon afterwards by Crampin and Hilton (as reported in [Kl82]), who proved that in fact, for all large \(X\), \(\lvert A\cap [1,X]\rvert \ll X^{\tau+o(1)}\) where \(\tau\approx 0.900626\) is the unique positive root of \[6^{-\tau}+\sum_{k\geq 0}(3\cdot 2^k)^{-\tau}=1.\] Their proo