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Statement

The test function (5) of a subdivision is defined on the Theorem 1 page: $f(x)=1$ on the lower halves $x_i\le x<\frac12(x_i+x_{i+1})$ of the intervals and $f(x)=-1$ on the upper halves.

Lemma 1 (printed pp. 138--139, quoted). "Let $N>1$, $\varepsilon>0$. There is a subdivision

$$0=x_0<x_1<\ldots<x_h=1$$

of the unit interval with

$$x_{i+1}-x_i<\varepsilon\qquad(i=0,1,\ldots,h-1) \tag{10}$$

and with the following property. Define $f(x)$ in $0\leqq x<1$ by (5). There is a subset $\sigma_\varepsilon$ of the unit interval of measure less than $\varepsilon$ such that

$$f(x)=f(mx) \tag{11}$$

if $x$, $mx$ are in the unit interval but $x\notin\sigma_\varepsilon$, and if $m$ is an integer with $1\leqq m\leqq N$."

Source. W. M. Schmidt, Disproof of some conjectures on Diophantine approximations, Studia Sci. Math. Hungar. 4 (1969), 137--144; Lemma 1 begins on printed p. 138 and ends on p. 139, its proof runs from p. 139 to p. 140, read on the page images. The edition read is identified on the source card.

Read depth. Claims checked: the statement was read clause by clause on the page images of pp. 138--139; the proof (pp. 139--140) was read for its structure and not checked. Nothing here is independently reviewed.

Proof pointer

Pages 139--140. Dirichlet's theorem on simultaneous approximation gives arbitrarily large $q$ and integers $p_m$ with $|\log m-p_m/q|<q^{-1-1/N}$ for $1\le m\le N$ (12). The auxiliary function $f^*$ on $0<x<1$ uses the breakpoints $e^{t/q}$, $t$ a negative integer, so that in the variable $\log x$ it is periodic with period $1/q$ (13); multiplication by $m$ shifts $\log x$ by $p_m/q$ up to an error below $q^{-1-1/N}$, so $f^*(x)=f^*(mx)$ outside a set $\sigma(q)$ of measure $\ll q^{-1/N}$. The subdivision (14) is $x_1=e^{-q}$, $x_{j}=e^{-q+(j-1)/q}$, up to $x_{q^2+1}=x_h=1$, with $f$ equal to $f^*$ above $e^{-q}$; its mesh is $\ll1/q$, and $q>q_0(\varepsilon)$ gives the lemma. Not reconstructed here.

Dependencies

Dirichlet's theorem on simultaneous approximation.

Bears on

• Problem 492: the lemma is the building block of Theorem 1, whose proof (Section 3, pp. 140--141) rescales the lemma's subdivisions to the blocks $[M_k,N_k)$ with $N=N_k$ and $\varepsilon=\varepsilon_k$. The lemma alone settles nothing about the problem.