Borderline near-linear growth rates

Determine the Hausdorff dimension of \(E_Q\) for sequences \(Q=(Q_m)_{m\ge0}\) satisfying \(Q_{m+1}/Q_m\to\infty\) while \(\log Q_{m+1}/\log Q_m\to1\), including the model growth rate \(Q_{m+1}\asymp Q_m(\log Q_m)^\lambda\) for \(\lambda>0\).

Background

The paper settles the dimension for power-type growth Qm+1≍QmτQ_{m+1}\asymp Q_m^\tau: bounded-ratio or linear-logarithmic growth gives dimension zero when τ=1\tau=1, whereas any τ>1\tau>1 gives dimension two. It identifies a remaining borderline regime in which the ratios diverge but the logarithmic growth exponent tends to one. The authors state that these cases remain unsolved, with Qm+1≍Qm(log⁡Qm)λQ_{m+1}\asymp Q_m(\log Q_m)^\lambda offered as a natural example.

References

An important class of remaining unsolved cases consists of sequences for which $$ \frac{Q_{m+1}{Q_m}\to +\infty\quad\text{but}\quad \frac{\log{Q_{m+1}{\log{Q_m}\to 1. $$ For example, a natural growth rate of $Q=(Q_m){m\geq 0}$ satisfying E:lograte is $$ Q_m\nearrow +\infty\quad\text{and}\quad Q{m+1}\asymp Q_m(\log{Q_m}){\lambda} \text{ for some }\lambda>0.

E:lograte:

Qm+1Qm→+∞butlog⁡Qm+1log⁡Qm→1.\frac{Q_{m+1}}{Q_m}\to +\infty\quad\text{but}\quad \frac{\log{Q_{m+1}}}{\log{Q_m}}\to 1.

— Winning property of counterexamples to Uniform Littlewood's Conjecture  (2608.24401 - Wu et al., 25 Aug 2026) in Section 6, paragraph following the summary of answers to Problem 1+epsilon