The logarithmic threshold and a malformed epsilon quantifier

Definitions and mathematical statement

Write $\|x\|=\min_{p\in{\mathbb{Z}}}|x-p|$ and take integral denominators $q\to\infty$. The literal Question 27, attributed to A. Pollington, asks for $\alpha,\beta\in{\mathbb{R}}$ such that \[ \text{for every }\varepsilon>0,\qquad \liminf_{q\to\infty}q(\log q)^{2-\varepsilon} \|q\alpha\|\|q\beta\|>0. \] The aid reports that the official statement really has this quantifier. It is false literally because it includes $\varepsilon>2$. If a coordinate is rational, its denominator multiples give zero. Otherwise its continued-fraction denominators satisfy $q_n\|q_n\alpha\|<1$; since $\|q_n\beta\|\leq1/2$, any exponent $s=2-\varepsilon<0$ gives a subsequence bounded by $\frac12(\log q_n)^s\to0$. The nonnegative liminf is therefore zero for every pair at each negative exponent.

Two repaired scopes must be distinguished. A near-$2$ reading asks only for all sufficiently small positive $\varepsilon$, hence exponents $s$ in some interval immediately below $2$. The stronger reading restricts the original quantifier to $0<\varepsilon<2$, equivalently asks for every fixed $s>0$ after using monotonicity for larger logarithmic powers. The aid credits Badziahin with a full-Hausdorff-dimensional family having an eventual bound \[ q\log q\log\log q\,\|q\alpha\|\|q\beta\|\geq c>0 \] for sufficiently large $q$. For those same pairs every fixed $s>1$ gives a positive, in fact infinite, liminf with weight $q(\log q)^s$. Thus the near-$2$ version is covered by the reported theorem, whereas the all-$s>0$ version is beyond the supplied known results. No claim of existence at every $0<s\leq1$ is inferred.

The aid gives a necessary continued-fraction constraint for the strong version. If, for one $s>0$, there are $c_s>0,Q_s$ such that \[ q(\log q)^s\|q\alpha\|\|q\beta\|\geq c_s \quad(q\geq Q_s), \] then both coordinates are irrational. At consecutive convergent denominators of either coordinate, denoted here by $q_n,q_{n+1}$, one has eventually \[ \frac{q_{n+1}}{q_n}<\frac{(\log q_n)^s}{2c_s},\qquad a_{n+1}<\frac{(\log q_n)^s}{2c_s}, \] where $a_{n+1}$ is the next partial quotient. This uses $\|q_n\alpha\|<1/q_{n+1}$ and the other coordinate's bound $1/2$, with the roles exchanged for the other convergent sequence. If a pair satisfies every $s>0$, these inequalities force $a_{n+1}=(\log q_n)^{o(1)}$ for each coordinate separately. Constants and thresholds may depend on $s$; the condition is necessary, not sufficient for the pairwise multiplicative lower bound.

The source additionally cites [65] for the almost-everywhere identity \[ \liminf_{q\to\infty}q(\log q)^2\|q\alpha\|\|q\beta\|=0. \] Its next sentence is truncated. The literal refutation, the near-$2$ repaired result, and the stronger unresolved interpretation have different quantifiers and are not combined into one status.

Short English statement

The printed "every $\varepsilon>0$" condition is impossible, since it includes negative logarithmic powers. The supplied aid distinguishes a near-$2$ repair, covered by Badziahin's reported bound, from the stronger demand for every positive logarithmic power. It also gives a necessary sub-power-of-log partial-quotient condition for that stronger demand.

Sources

• Local UnsolvedMath registry, permanent ID $20003302$, source code catalogue entry; canonical location aim-other-notes.json, notes[108].
• AIM Emerging applications of measure rigidity, Question 27 (A. Pollington), https://aimath.org/WWN/measrigid/measrigid.pdf. The supplied verification says the official mathematics is unambiguous while the assigned extraction ends mid-sentence after "Thus, the set of $(\alpha,\beta)$ in"; its registry statement is reconstructed/unverified.
• Source label [65] for the exponent-$2$ almost-everywhere zero liminf; no full entry is supplied. Badziahin's full-dimensional $q\log q\log\log q$ lower-bound theorem is named in the attached summary without a full citation.
• Research attempt 1 is marked invalid statement/refuted, proved, and low novelty confidence, despite the registry open label. Its explicit caveat says the aid is machine generated, not peer reviewed, and requires independent verification before citation.