Erdős Problem #994

Erdős Problem 994: a fixed measurable counterexample

Statement and attribution

The intended Khintchine conjecture asks whether, for each fixed measurable $E\subset(0,1)$, the averages $N^{-1}\sum_{n\le N}1_E(\{n\alpha\})$ converge to $\lambda(E)$ for almost every $\alpha$. This is false. We reconstruct a counterexample through the finite transference argument of Quas--Wierdl, Rates of divergence of nonconventional ergodic averages, Section 5, which gives another route to Marstrand's 1970 result. The transference lemma below is an explicit external input, obtained from the multiparameter Rokhlin tower theorem; the small-set construction and its assembly are proved here. No novelty is claimed.

The stronger literal order for almost all $\alpha$, for all $E$ fails trivially: for each irrational $\alpha$ take its countable orbit as $E$. That observation does not produce a single counterexample set independent of $\alpha$, and is not the argument used below.

The precise finite transference input

Here is the indicator-function form of the copying argument in the proof of Quas--Wierdl Lemma 5.1. Let finitely many positive integers $n$ be specified, and let the primes dividing them be $p_1,\ldots,p_d$. Choose $M>0$ such that $M,\log p_1,\ldots,\log p_d$ are linearly independent over $\mathbb Q$. For a measurable $H\subset\mathbb R/M\mathbb Z$, finitely many averages over those integers, and $\eta>0$, there exists measurable $E\subset[0,1)$ such that \[ \lambda(E)\le |H|/M+\eta, \] and every lower bound for the normalized measure of a joint event involving those averages of $1_H(y-\log n\bmod M)$ transfers to the corresponding averages of $1_E(nx\bmod1)$ with a loss of at most $\eta$.

The relevant point is that this input preserves indicators, not just arbitrary $L^p$ functions. In the source proof, the commuting actions $x\mapsto p_ix\bmod1$ and $y\mapsto y-\log p_i\bmod M$ are represented on finite Rokhlin towers of the same rectangular shape. Copying the values $0,1$ level by level gives a $0,1$ function. For any fixed finite set of times, the proportion of levels whose required translates leave the tower tends to zero as its side length grows; the uncovered error sets also tend to zero. On all other levels the specified averages agree exactly. This explains the stated finite-event form and the two error terms. The existence of these towers is the external part, as in Lemma 5.1.

A small set causing a large maximal average

Fix integers $J\ge3$ and $K\ge\max(12,2J+4)$. Choose \[ (J+K+1)\log2<M<(J+K+2)\log2 \] outside the countable rational span of the logarithms of primes at most $2^{J+K}$. Set $H=[0,2\log2)$ in the circle of circumference $M$. For $J\le j<J+K$ and $y\in[j\log2,(j+1)\log2)$, all integers \[ \lceil e^y/4\rceil\le n\le2^j \] satisfy $0\le y-\log n\le2\log2$, apart from irrelevant endpoints. There are at least $2^{j-1}-1\ge3\cdot2^{j-3}$ of them. Therefore \[ \frac1{2^j}\sum_{n\le2^j}1_H(y-\log n\bmod M)\ge\frac38. \] The union of these $K$ disjoint $y$-intervals has normalized measure $K\log2/M\ge2/3$, while $|H|/M\le2/K$. Apply finite indicator transference with $\eta=1/K$ and the $K$ averages just specified. We obtain a measurable $E_{J,K}\subset[0,1)$ with \[ \lambda(E_{J,K})\le3/K,\qquad \lambda\left\{x:\max_{J\le j<J+K} \frac1{2^j}\sum_{n\le2^j}1_{E_{J,K}}(\{nx\})\ge3/8\right\}\ge1/2. \tag{1} \] Only finitely many times occur in each application, which is essential to the transference step.

One fixed set, infinitely many late failures

For $m\ge1$, take $J_m=m+3$ and choose an integer $K_m$ large enough for $1$ and for $3/K_m\le2^{-m-5}$. Let $E_m=E_{J_m,K_m}$ and set $E=(\bigcup_{m\ge1}E_m)\setminus\{0\}$. Then $E\subset(0,1)$ and $\lambda(E)\le1/32$. Let $B_m$ be the event in $1$. Its measure is at least $1/2$, so \[ \lambda(\limsup_mB_m) =\lim_{r\to\infty}\lambda\left(\bigcup_{m\ge r}B_m\right)\ge1/2. \] No independence of these events is needed. For every irrational $x$ in this limsup, infinitely many $m$ supply an index $N=2^j\ge2^{J_m}$ for which the average of $1_{E_m}(\{nx\})$ is at least $3/8$. The same is true for $E$, since $E_m\setminus\{0\}\subset E$ and an irrational orbit never visits $0$. These indices tend to infinity. Hence on a set of measure at least $1/2$, \[ \limsup_{N\to\infty}\frac1N\sum_{n\le N}1_E(\{nx\})\ge\frac38>\lambda(E). \] This disproves the intended fixed-set conjecture.

Conclusion and external-dependency scope

The counterexample is complete relative to the stated finite Rokhlin transference input. A proof of the underlying multiparameter tower theorem is not supplied, and no claim of an elementary proof independent of ergodic theory is made. Sources: https://www.erdosproblems.com/994; J. M. Marstrand, https://doi.org/10.1112/plms/s3-21.3.540; A. Quas and M. Wierdl, Section 5 and the proof of Lemma 5.1, https://web.uvic.ca/~aquas/papers/paper24.pdf.

COMPLETION ESTIMATE

COMPLETION: 100%