Erdős Problem #495

Erdős Problem 495 --- collection record

PROVENANCE AND STATUS

Official problem: https://www.erdosproblems.com/495

Saved collaborative-database status: open. Snapshot checked: 2026-09-23T17:22:24Z. Latest status and formalization links: https://teorth.github.io/erdosproblems/

This is an attributed collection of existing writeups. The model-folder names below identify their original attribution; placement in GPT_6_Astra_Ultra does not attribute their mathematical work to that model. All locally available source versions selected for this problem are reproduced below, without editing their text.

Proof claims, computations, literature statements, and completion estimates in the reused text have not been independently verified in this collection pass. A source claiming a solution does not change the saved upstream status.

Primary source (latest numbered version in the preferred model folder): attacks/erdos/gpt_pro_5.2/495.tex.

REUSED SOURCE 1

Original attribution: gpt_pro_5.2. Original file: attacks/erdos/gpt_pro_5.2/495.tex.

FORMAL RESTATEMENT

For a real number $x$, define $\|x\|:=\min_{m\in\mathbb{Z}}|x-m|$, the distance from $x$ to the nearest integer. Given $\alpha,\beta\in\mathbb{R}$, the Littlewood conjecture asserts: \[ \liminf_{n\to\infty} n\,\|n\alpha\|\,\|n\beta\| = 0. \] Question: Is this statement true for all real pairs $(\alpha,\beta)$?

Edge cases: if $\alpha\in\mathbb{Q}$ then $\|n\alpha\|=0$ for infinitely many $n$, so the liminf is $0$ (similarly for $\beta\in\mathbb{Q}$).

QUICK LITERATURE/CONTEXT CHECK

The statement identifies this as the (infamous) Littlewood conjecture. No additional background is provided in the problem file, and per the integrity rule I do not cite or rely on external results here.

ATTACK PLAN

Proof track:

• Prove the conjecture in trivial/degenerate cases (e.g. rational parameters).
• Establish unconditional bounds from classical Diophantine approximation (Dirichlet) to at least show $\liminf$ is finite and small.

Disproof track:

• Attempt to construct a pair $(\alpha,\beta)$ with a positive lower bound on $n\|n\alpha\|\|n\beta\|$; no such construction is known to me and would be highly nontrivial.

I provide two proved lemmas and numerical sanity checks, but no full proof/disproof.

WORK

Lemma 1 (rational parameter case).

If $\alpha\in\mathbb{Q}$ or $\beta\in\mathbb{Q}$, then \[\liminf_{n\to\infty} n\,\|n\alpha\|\,\|n\beta\| = 0. \]

Proof.

Assume $\alpha\in\mathbb{Q}$. Then $\alpha=p/q$ with integers $p,q\ge 1$ in lowest terms. For every multiple $n=tq$, we have $n\alpha=tp\in\mathbb{Z}$, hence $\|n\alpha\|=0$. Therefore for infinitely many $n$ the product $n\|n\alpha\|\|n\beta\|$ equals $0$, so the liminf is $0$. The argument with $\beta\in\mathbb{Q}$ is identical. $\square$

Lemma 2 (Dirichlet simultaneous approximation $\Rightarrow$ universal bound $\liminf\le 1$).

For every pair $(\alpha,\beta)\in\mathbb{R}^2$, there exist infinitely many integers $n\ge 1$ such that \[n\,\|n\alpha\|\,\|n\beta\| \le 1. \] In particular, \[\liminf_{n\to\infty} n\,\|n\alpha\|\,\|n\beta\| \le 1. \]

Proof.

Fix an integer $N\ge 1$. Consider the $N+1$ points in the unit square $[0,1)^2$ given by \[P_j := (\{j\alpha\},\{j\beta\})\quad\text{for }j=0,1,\dots,N, \] where $\{x\}=x-\lfloor x\rfloor$ denotes the fractional part. Partition $[0,1)^2$ into a grid of $\lceil\sqrt N\rceil\times\lceil\sqrt N\rceil$ rectangles, each of side length at most $1/\sqrt N$. There are at most $N$ such rectangles. By the pigeonhole principle, two distinct points $P_i,P_j$ with $0\le i<j\le N$ fall in the same rectangle. Let $n=j-i$ (so $1\le n\le N$). Then the coordinate differences satisfy \[\|n\alpha\| \le |\{j\alpha\}-\{i\alpha\}| \le \frac{1}{\sqrt N},\qquad \|n\beta\| \le |\{j\beta\}-\{i\beta\}| \le \frac{1}{\sqrt N}. \] (The first inequality uses that the distance to the nearest integer is at most the distance between two fractional parts; if the difference crosses $0$, one can instead use $1-|\{j\alpha\}-\{i\alpha\}|$, which is also bounded by the rectangle size. The same holds for $\beta$.) Therefore \[n\,\|n\alpha\|\,\|n\beta\| \le N\cdot \frac{1}{\sqrt N}\cdot\frac{1}{\sqrt N}=1. \] Since this holds for each $N$, and produces some $1\le n\le N$, taking $N\to\infty$ yields infinitely many such $n$. $\square$

FAST REALITY CHECK (numerical search for sample pairs).

For two sample pairs I computed \[\min_{1\le n\le N} n\,\|n\alpha\|\,\|n\beta\|\] for $N\in\{10,100,1000,10000\}$:

• $(\alpha,\beta)=(\sqrt2,\sqrt3)$: minimum values were $0.0874886095$ at $N=10$ (attained at $n=7$), $0.00995678225$ at $N=100$ (attained at $n=41$), and unchanged through $N=10000$.
• $(\alpha,\beta)=(\sqrt2,\pi)$: minimum values were $0.00622729112$ at $N=10$ (attained at $n=7$), unchanged at $N=100$, then $0.000660373015$ at $N=1000$ (attained at $n=113$), unchanged through $N=10000$.

These are only sanity checks (they do not prove convergence of the liminf).

VERIFICATION

• Lemma 1: verified that $\|n\alpha\|=0$ along multiples of the denominator, forcing liminf $0$.
• Lemma 2: verified pigeonhole step in $[0,1)^2$ and the conversion from closeness of fractional parts to bounds on $\|n\alpha\|$ and $\|n\beta\|$.
• Numerical check: confirmed distances computed as $|x-\mathrm{round}(x)|$.

FINAL

UNRESOLVED

(i) Strongest proved partial result.

• If $\alpha$ or $\beta$ is rational, the conjectured liminf equals $0$.
• For all $(\alpha,\beta)$, Dirichlet's simultaneous approximation yields infinitely many $n$ with $n\|n\alpha\|\|n\beta\|\le 1$, so the liminf is always $\le 1$.

(ii) First gap (crisp). Improve the universal bound from Lemma 2 to show that for every pair $(\alpha,\beta)$ one can achieve \[n\,\|n\alpha\|\,\|n\beta\| < \varepsilon\] for arbitrarily small $\varepsilon>0$ (equivalently, show the liminf is $0$).

(iii) Top 3 next moves.

1. Strengthen the pigeonhole argument using finer partitions or geometric methods (e.g. looking for lattice points in regions defined by $\|n\alpha\|\,\|n\beta\|$) to push the bound below an absolute constant.
2. Analyze special classes of pairs $(\alpha,\beta)$ (e.g. one parameter with exceptionally good rational approximations) to force $n\|n\alpha\|\to 0$ along a subsequence, which would imply the conjecture for that pair.
3. Computationally search for sequences $n$ minimizing $n\|n\alpha\|\|n\beta\|$ for specific pairs (e.g. quadratic irrationals) to guess structure of near-minimizers.

(iv) Minimal counterexample structure. A counterexample would be a pair $(\alpha,\beta)$ and a constant $c>0$ such that \[n\,\|n\alpha\|\,\|n\beta\|\ge c\quad\text{for all }n\ge 1. \] Such a pair would have to simultaneously avoid very small fractional parts in both coordinates across all integer multiples, exhibiting a strong form of "simultaneous bad approximation".

COMPLETION ESTIMATE

The primary source supplies no extractable numerical completion estimate. Its mathematical progress is not numerically assessed here.

Newly verified progress in this collection pass: $0\%$. This measures new verification work, not the amount of mathematics achieved in the reused text.

FORMAL RESTATEMENT

For a real number $x$, define $\|x\|:=\min_{m\in\mathbb{Z}}|x-m|$, the distance from $x$ to the nearest integer. Given $\alpha,\beta\in\mathbb{R}$, the Littlewood conjecture asserts: \[ \liminf_{n\to\infty} n\,\|n\alpha\|\,\|n\beta\| = 0. \] Question: Is this statement true for all real pairs $(\alpha,\beta)$?

Edge cases: if $\alpha\in\mathbb{Q}$ then $\|n\alpha\|=0$ for infinitely many $n$, so the liminf is $0$ (similarly for $\beta\in\mathbb{Q}$).

QUICK LITERATURE/CONTEXT CHECK

The statement identifies this as the (infamous) Littlewood conjecture. No additional background is provided in the problem file, and per the integrity rule I do not cite or rely on external results here.

ATTACK PLAN

Proof track:

• Prove the conjecture in trivial/degenerate cases (e.g. rational parameters).
• Establish unconditional bounds from classical Diophantine approximation (Dirichlet) to at least show $\liminf$ is finite and small.

Disproof track:

• Attempt to construct a pair $(\alpha,\beta)$ with a positive lower bound on $n\|n\alpha\|\|n\beta\|$; no such construction is known to me and would be highly nontrivial.

I provide two proved lemmas and numerical sanity checks, but no full proof/disproof.

WORK

Lemma 1 (rational parameter case).

If $\alpha\in\mathbb{Q}$ or $\beta\in\mathbb{Q}$, then \[\liminf_{n\to\infty} n\,\|n\alpha\|\,\|n\beta\| = 0. \]

Proof.

Assume $\alpha\in\mathbb{Q}$. Then $\alpha=p/q$ with integers $p,q\ge 1$ in lowest terms. For every multiple $n=tq$, we have $n\alpha=tp\in\mathbb{Z}$, hence $\|n\alpha\|=0$. Therefore for infinitely many $n$ the product $n\|n\alpha\|\|n\beta\|$ equals $0$, so the liminf is $0$. The argument with $\beta\in\mathbb{Q}$ is identical. $\square$

Lemma 2 (Dirichlet simultaneous approximation $\Rightarrow$ universal bound $\liminf\le 1$).

For every pair $(\alpha,\beta)\in\mathbb{R}^2$, there exist infinitely many integers $n\ge 1$ such that \[n\,\|n\alpha\|\,\|n\beta\| \le 1. \] In particular, \[\liminf_{n\to\infty} n\,\|n\alpha\|\,\|n\beta\| \le 1. \]

Proof.

Fix an integer $N\ge 1$. Consider the $N+1$ points in the unit square $[0,1)^2$ given by \[P_j := (\{j\alpha\},\{j\beta\})\quad\text{for }j=0,1,\dots,N, \] where $\{x\}=x-\lfloor x\rfloor$ denotes the fractional part. Partition $[0,1)^2$ into a grid of $\lceil\sqrt N\rceil\times\lceil\sqrt N\rceil$ rectangles, each of side length at most $1/\sqrt N$. There are at most $N$ such rectangles. By the pigeonhole principle, two distinct points $P_i,P_j$ with $0\le i<j\le N$ fall in the same rectangle. Let $n=j-i$ (so $1\le n\le N$). Then the coordinate differences satisfy \[\|n\alpha\| \le |\{j\alpha\}-\{i\alpha\}| \le \frac{1}{\sqrt N},\qquad \|n\beta\| \le |\{j\beta\}-\{i\beta\}| \le \frac{1}{\sqrt N}. \] (The first inequality uses that the distance to the nearest integer is at most the distance between two fractional parts; if the difference crosses $0$, one can instead use $1-|\{j\alpha\}-\{i\alpha\}|$, which is also bounded by the rectangle size. The same holds for $\beta$.) Therefore \[n\,\|n\alpha\|\,\|n\beta\| \le N\cdot \frac{1}{\sqrt N}\cdot\frac{1}{\sqrt N}=1. \] Since this holds for each $N$, and produces some $1\le n\le N$, taking $N\to\infty$ yields infinitely many such $n$. $\square$

FAST REALITY CHECK (numerical search for sample pairs).

For two sample pairs I computed \[\min_{1\le n\le N} n\,\|n\alpha\|\,\|n\beta\|\] for $N\in\{10,100,1000,10000\}$:

• $(\alpha,\beta)=(\sqrt2,\sqrt3)$: minimum values were $0.0874886095$ at $N=10$ (attained at $n=7$), $0.00995678225$ at $N=100$ (attained at $n=41$), and unchanged through $N=10000$.
• $(\alpha,\beta)=(\sqrt2,\pi)$: minimum values were $0.00622729112$ at $N=10$ (attained at $n=7$), unchanged at $N=100$, then $0.000660373015$ at $N=1000$ (attained at $n=113$), unchanged through $N=10000$.

These are only sanity checks (they do not prove convergence of the liminf).

VERIFICATION

• Lemma 1: verified that $\|n\alpha\|=0$ along multiples of the denominator, forcing liminf $0$.
• Lemma 2: verified pigeonhole step in $[0,1)^2$ and the conversion from closeness of fractional parts to bounds on $\|n\alpha\|$ and $\|n\beta\|$.
• Numerical check: confirmed distances computed as $|x-\mathrm{round}(x)|$.

FINAL

UNRESOLVED

(i) Strongest proved partial result.

• If $\alpha$ or $\beta$ is rational, the conjectured liminf equals $0$.
• For all $(\alpha,\beta)$, Dirichlet's simultaneous approximation yields infinitely many $n$ with $n\|n\alpha\|\|n\beta\|\le 1$, so the liminf is always $\le 1$.

(ii) First gap (crisp). Improve the universal bound from Lemma 2 to show that for every pair $(\alpha,\beta)$ one can achieve \[n\,\|n\alpha\|\,\|n\beta\| < \varepsilon\] for arbitrarily small $\varepsilon>0$ (equivalently, show the liminf is $0$).

(iii) Top 3 next moves.

1. Strengthen the pigeonhole argument using finer partitions or geometric methods (e.g. looking for lattice points in regions defined by $\|n\alpha\|\,\|n\beta\|$) to push the bound below an absolute constant.
2. Analyze special classes of pairs $(\alpha,\beta)$ (e.g. one parameter with exceptionally good rational approximations) to force $n\|n\alpha\|\to 0$ along a subsequence, which would imply the conjecture for that pair.
3. Computationally search for sequences $n$ minimizing $n\|n\alpha\|\|n\beta\|$ for specific pairs (e.g. quadratic irrationals) to guess structure of near-minimizers.

(iv) Minimal counterexample structure. A counterexample would be a pair $(\alpha,\beta)$ and a constant $c>0$ such that \[n\,\|n\alpha\|\,\|n\beta\|\ge c\quad\text{for all }n\ge 1. \] Such a pair would have to simultaneously avoid very small fractional parts in both coordinates across all integer multiples, exhibiting a strong form of "simultaneous bad approximation".