Let alpha,beta ∈ R. Is it true that liminf_n→ ∞ n | n alpha | | n beta| =0 where |x| is the distance from x to the nearest integer?

Erdős Problem #495 — Let $\alpha,\beta \in \mathbb{R}$ . Is it true that $\liminf_{n\to \infty} n \| n\alpha \| \| n\beta\| =0$ where $\|x\|$ is the distance from $x$ to the nearest integer?

About 65 years old · traced to 1961

Let $\alpha,\beta \in \mathbb{R}$ . Is it true that $\liminf_{n\to \infty} n \| n\alpha \| \| n\beta\| =0$ where $\|x\|$ is the distance from $x$ to the nearest integer?

References

Primary source

ULAM AI, UnsolvedMath dataset, Erdős Problems set, revision b9437975f3c873f635a13c48f8b022f5ba80898a; CC BY 4.0.

Additional references

UnsolvedMath, Erdős Problems set, ULAM AI, licensed CC BY 4.0.

diophantine-approximation number-theory

InErdős Problems

Posted 1 month ago · edited 3 weeks ago

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