Franklin's Notes


Distribution of logarithm modulo 1

A notable example of a naturally-ocurring sequence of real numbers that is not uniformly distributed mod 1 is the sequence $x_n=\log n$, the sequence of natural logarithms of the natural numbers. In this note I'd like to take a careful look at the fractional parts of this sequence. In particular, I'd like to calculate the values of in terms of $\alpha\in (0,1]$.

First of all, if $k\in\mathbb{N}$, notice that for all values of $n$ between $[e^k]$ and $[e^{k+\alpha}]-1$ inclusive, we have that $A([0,\alpha);n+1)=A([0,\alpha);n)+1$, since ${x_{n+1}}\in [0,\alpha)$. Further, these are the only values of $n$ for which $A([0,\alpha);n+1)=A([0,\alpha);n)+1$. This means that the ratio increases as $n$ increases to $n+1$ precisely for these values of $n$. Hence, the supremum of this sequence over all values of $n\in\mathbb{N}$ is equal to the supremum of the values that it takes at the subsequence $n=[e^{k+\alpha}]$, where $k$ ranges over $\mathbb{N}$.

The above observation, together with a simple counting argument, tells us that Now, since $|x-[x]|<1$ for all $x\in\mathbb{R}$, we have that or, simplifying the geometric sum, which means that which gives us the value of the desired limit at last: This is already sufficient to conclude that the sequence $x_n=\log n$ is not uniformly distributed modulo $1$, since uniform distribution would require that this limit equal $\alpha$.

Now we calculate the value of the infimum density. By a similar argument as before, the value of $A([0,\alpha),n)$ fails to increase when $n$ increases to $n+1$ precisely for the values of $n$ between $[e^{k+\alpha}]$ and $[e^{k+1}]-1$ for integers $k$. This means that the ratio decreases as $n\mapsto n+1$ precisely for these values of $n$, meaning that its limit infimum as $n\to\infty$ is equal to the limit infimum of the subsequence with $n=[e^{k+1}]$ for $k\in\mathbb{N}$.

This time, we have that and, following similar algebra as before, we have and finally we have the desired result

By similar reasoning, we can show that for the sequence $x_n=\log_b n$ for $b>1$, the following formulas hold: This means that, for larger bases $b$, there is a greater discrepancy between the supremum and infimum densities of the points ${x_n}_{n\leq N}$ as $N\to\infty$ - that is, the densities have a more pronounced oscillation for larger bases $b$. These facts are related to Benford's Law, which is an observation about the distribution of the leading significant digits of random data.

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