Does $\frac{1}{\log n}\sum_{k\le n}(\frac12-\{\alpha k\})$ have a limiting distribution in $\alpha$? (Erdős #1002)
Statement
For $0<\alpha<1$, let $$f(\alpha,n)=\frac{1}{\log n}\sum_{1\leq k\leq n}\left(\tfrac{1}{2}-\{ \alpha k\}\right),$$ where $\{x\}$ denotes the fractional part of $x$. Does $f(\alpha,n)$ have an asymptotic distribution function? In other words, is there a non-decreasing function $g$ with $g(-\infty)=0$ and $g(\infty)=1$ such that $$\lim_{n\to \infty}\lvert \{ \alpha\in (0,1): f(\alpha,n)\leq c\}\rvert=g(c),$$ where $\lvert\cdot\rvert$ denotes Lebesgue measure?
Acceptance. FULLY RESOLVES: a complete proof that $f(\alpha,n)$ has an asymptotic distribution function — ideally identifying $g$ (e.g. a Kesten-type Cauchy law) — or a proof that no asymptotic distribution function exists. Machine-checkable proof preferred (the statement is already formalised in the formal-conjectures Lean repository), else a complete written proof with all steps. ADVANCES: existence of the limiting distribution along explicit subsequences of $n$ or after restricting/conditioning on structured classes of $\alpha$, stated and proved precisely; identification of the limit of the characteristic functions or moments of $f(\cdot,n)$; tightness or non-degeneracy theorems for the family $\{f(\cdot,n)\}_n$; or a large-scale reproducible numerical study that evaluates $f(\alpha,n)$ exactly via the continued-fraction/Ostrowski representation (not floating-point summation over $k$) and quantifies convergence or non-convergence of the empirical distribution functions at rapidly growing $n$ — deliver code, data, and empirical-CDF distance tables. Deliver the proof file, or the code plus data and analysis for the computational milestone.
Background
Posed by Erdős [Er64b]; listed as open on erdosproblems.com/1002 (fetched 2026-07-13, status 'open', tagged 'analysis | diophantine approximation'). The sum $\sum_{k\leq n}(\tfrac12-\{\alpha k\})$ is the classical centred equidistribution error of the sequence $\alpha k \bmod 1$, whose behaviour is governed by the continued-fraction expansion of $\alpha$. The frontier result is Kesten's theorem [Ke60]: for the shifted sums $$f(\alpha,\beta,n)=\frac{1}{\log n}\sum_{1\leq k\leq n}\left(\tfrac{1}{2}-\{\beta+\alpha k\}\right),$$ with both $\alpha$ and the shift $\beta$ averaged, the asymptotic distribution function exists and is a Cauchy law: $g(c)=\frac{1}{\pi}\int_{-\infty}^{\rho c}\frac{1}{1+t^2}\,\mathrm{d}t$ for an explicit constant $\rho>0$. Erdős's question is the unshifted case $\beta=0$, where the extra averaging that powers Kesten's argument is unavailable and even the existence of a limiting distribution is unknown. The statement has been formalised in Lean in Google DeepMind's formal-conjectures repository (linked from the problem page). The attacker's tool: the Ostrowski/continued-fraction representation of the sawtooth sums, which enables exact and fast evaluation of $f(\alpha,n)$ for astronomically large $n$ — supporting a rigorous-arithmetic empirical-distribution study across $\alpha$ — and is also the natural route for adapting Kesten's Fourier-analytic/ergodic machinery to the $\beta=0$ case.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #1002 (T. F. Bloom) | website |
| REF-02 | Formalised statement of Erdős #1002 (google-deepmind/formal-conjectures, Lean 4) | website |
Investigations · 0
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