请求验证π的数字均匀性证明的正确性

Hey there! Thanks for sharing your proof attempt—totally get the urge to dive into this question, even if it turns out to be an open problem. Let's walk through the issues with your reasoning step by step:

Key Problems in the Proof #

1. Circular Reasoning in Probability Assumptions #

Your proof starts by assuming each digit $\mathcal{D}_i$ follows a Bernoulli distribution with fixed probability $p_l$. But this is exactly what we're trying to prove when we say π's digits are uniform (uniformity would mean $p_l = 1/10$ for all $l$). By presupposing fixed probabilities exist, you're skipping the core question: do π's digits behave like independent, identically distributed uniform random variables? We don't know that yet—it's the conjecture we're trying to verify.

2. Misapplication of Borel-Cantelli Lemma #

The second Borel-Cantelli lemma tells us that if $p_l > 0$, then the number of times digit $l$ appears ($n_l$) will be infinite almost surely. That part is correct, but it doesn't lead to the conclusion that all $n_l$ are equal. Even if different digits had different positive probabilities, each would still appear infinitely often. The lemma doesn't say anything about the relative frequencies of the digits converging to the same value—that's a job for the Strong Law of Large Numbers, which requires the trials to be iid with a known mean (again, something we can't assume for π's digits).

3. Incorrect Claim About Equal Counts #

Your statement that $\Pr(n_0 = n_1 = \cdots = n_9) = 1$ is not valid. Even for a true uniform random sequence, the counts of each digit will never be exactly equal for all $l$ as we take more digits (though their relative frequencies will converge to $1/10$ almost surely). For an infinite sequence, "equality of counts" isn't a meaningful statement because all counts are infinite—we have to talk about the limit of $n_l / N$ as $N \to \infty$, not the absolute counts.

4. Notation Confusion #

Your definition of $\Pi$ is a bit muddled: you describe it as the "ordered set of π's digits" but then define it as a collection of subsets $S_l$ (each containing all instances of a single digit). That's a partition of the digits, not the ordered sequence itself. It's a minor point, but clarity here helps avoid misinterpretation.

What We Actually Know #

The uniformity of π's digits (a property called normality) is still an open problem. We have massive numerical evidence—trillions of digits have been computed, and their relative frequencies are extremely close to $1/10$ for each digit—but no formal proof exists that this holds for all digits.

Don't feel discouraged though! This is a classic example of a question that seems straightforward but touches on deep unsolved problems in number theory. It's awesome that you tried to tackle it head-on.

备注：内容来源于stack exchange，提问作者JerBear