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Statement

Subdivision and notation (§1, p. 757). A subdivision of $(0,\infty)$ is $\Delta=(z_0,z_1,\ldots)$ with $0=z_0<z_1<\cdots$ and $\lim_{n\to\infty}z_n=\infty$. For $z_{n-1}\le x<z_n$ the paper puts

$$[x]_\Delta=z_{n-1},\qquad \delta(x)=z_n-z_{n-1},\qquad \langle x\rangle_\Delta=\frac{x-[x]_\Delta}{\delta(x)},\qquad \phi(x)=n+\langle x\rangle_\Delta,$$

so that $0\le\langle x\rangle_\Delta<1$: $\delta(x)$ is the length of the interval of $\Delta$ containing $x$, and $\langle x\rangle_\Delta$ is the relative position of $x$ in it.

Definition (p. 757). Let $\{x_k\}$ be an increasing sequence of positive numbers. It is uniformly distributed modulo $\Delta$ (u.d. (mod $\Delta$)) when $\{\langle x_k\rangle_\Delta\}$ is uniformly distributed over $[0,1]$ in the sense that, for each $\alpha\in[0,1)$, the proportion of the numbers $\langle x_1\rangle_\Delta,\ldots,\langle x_k\rangle_\Delta$ lying in $[0,\alpha)$ tends to $\alpha$ as $k\to\infty$.

For the subdivision $\Delta_0$ with $z_n=n$ this is ordinary uniform distribution (mod 1), since then $[x]_\Delta=[x]$, $\delta(x)=1$ and $\langle x\rangle_\Delta$ is the fractional part of $x$. In general $\langle x_k\rangle_\Delta\equiv\phi(x_k)\pmod 1$ with $\phi$ a continuous polygonal function, so u.d. (mod $\Delta$) of $\{x_k\}$ is equivalent to u.d. (mod 1) of $\{\phi(x_k)\}$; the paper notes that $\phi$ need not be differentiable everywhere and that $\phi'$ is not monotonic unless $\delta(x)$ is assumed monotonic, which is why the classical criteria do not apply directly (p. 757).

Notation (p. 757). The arrows $\uparrow$, $\nearrow$, $\downarrow$, $\searrow$ mean increasing, non-decreasing, decreasing and non-increasing approach respectively.

Counting criterion (§2, p. 758). With $N(\alpha,x)$ the number of $k$ with $x_k\le x$ and $\langle x_k\rangle_\Delta<\alpha$, and $N(x)=N(1,x)$, the sequence $\{x_k\}$ is u.d. (mod $\Delta$) if and only if $\lim_{x\to\infty}N(\alpha,x)/N(x)=\alpha$ for each $\alpha\in[0,1)$.

Source. W. J. LeVeque, On uniform distribution modulo a subdivision, Pacific J. Math. 3 (1953), 757--771, §1 (printed p. 757) and the opening of §2 (printed p. 758), read on the page images. The edition read is identified in the source digest.

Proof pointer

A definition; the equivalence with u.d. (mod 1) of $\{\phi(x_k)\}$ and the counting criterion are stated without separate proof (pp. 757--758).

Dependencies

Ordinary uniform distribution (mod 1).

Bears on

• Problem 492: the problem's $f(x)=(x-a_i)/(a_{i+1}-a_i)$ for $x\in[a_i,a_{i+1})$ is $\langle x\rangle_\Delta$ for the subdivision with points $a_1<a_2<\cdots$ (for $x\ge a_1$), and the question asks whether $\{k\alpha\}$ is u.d. (mod $\Delta$) for almost all $\alpha>0$ in this sense (an authored identification; the paper does not pose the question).