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Statement

Setting (printed p. 138). H. T. Croft conjectured that for a set $S$ of positive reals of infinite Lebesgue measure, almost every $\alpha>0$ has infinitely many of its multiples $\alpha,2\alpha,3\alpha,\ldots$ in $S$; the weaker form asks only for some $\alpha$ with this property. Schmidt attributes the conjecture to a written communication from Croft to Erdős (footnote 1, citing Croft's mimeographed Research problems, Cambridge 1967, problem VII 8, p. 27). For a set $S$ of positive reals, $\mu_S(\varrho)$ denotes the measure of $S\cap(0,\varrho)$.

Theorem 2 (printed p. 138, quoted). "Let $\psi(\varrho)$ be a function satisfying $0<\psi(\varrho)<1$ which decreases to zero as $\varrho$ tends to infinity. There is an open set $S$ with

$$\mu_S(\varrho)\geqq\varrho\psi(\varrho) \tag{7}$$

such that for almost all $\alpha>0$, only finitely many of the numbers (8) $\alpha,2\alpha,3\alpha,\ldots$ lie in $S$. There is a measurable set $S^*$ with (7) such that for all $\alpha$, only finitely many of the numbers (8) are in $S^*$."

For any admissible $\psi$ with $\varrho\psi(\varrho)\to\infty$ (for instance $\psi(\varrho)=1/(2+\sqrt\varrho)$), (7) gives $S$ and $S^*$ infinite measure, so the open set $S$ refutes Croft's conjecture and $S^*$ refutes its weaker form. The paper's remarks on p. 138 add that the second assertion follows from the first by deleting from $S$ the points with infinitely many integral multiples in $S$ (Remark 1); that if $S$ is open and unbounded some $\alpha>0$ always has infinitely many multiples in $S$, citing Kingman's Corollary 1 of Theorem 1 (Remark 2); and that the sets constructed have $\mu_S(\varrho)=o(\varrho)$ (Remark 3), which Theorem 3 shows is necessary.

Source. W. M. Schmidt, Disproof of some conjectures on Diophantine approximations, Studia Sci. Math. Hungar. 4 (1969), 137--144; Theorem 2 and the remarks on printed p. 138, the proof in Section 4 on printed pp. 141--143, read on the page images. The edition read is identified on the source card.

Read depth. Claims checked: the statement, its setting and the remarks were read clause by clause on the page image of p. 138; the proof (pp. 141--143) was read for its structure and not checked. Nothing here is independently reviewed.

Proof pointer

Section 4 (pp. 141--143). For $0<N<M$, $0<\varepsilon<1$ and a positive integer $q$, $[N,M;q,\varepsilon]$ is the set of $x\in(N,M)$ lying in some interval $e^{t/q}<x<e^{(t+\varepsilon)/q}$ with $t$ an integer, and Lemma 2 (p. 142) bounds its measure by $\frac\varepsilon4(M-N)<\mu(N,M;q,\varepsilon)<3\varepsilon(M-N)$ when $M>2N$ and $q>32$. With $1=\varepsilon_1>\varepsilon_2>\cdots$ summable (21), integers $N_k>2N_{k-1}$ with $\psi(N_k)<\varepsilon_{k+1}/8$ (22), and $q_k$ from Dirichlet's theorem on simultaneous approximation of $\log m$, $1\le m\le N_k^2$ (23), $S$ is $(0,N_1)$ together with the sets $[N_{k-1},N_k;q_k,\varepsilon_k]$; Lemma 2 gives (7) (p. 143). For $\alpha$ in $b^{-1}\le\alpha\le b$, the set $S_k$ of $\alpha$ with a multiple in the $k$-th piece has $\mu(S_k)\ll\varepsilon_k$, by (23) and Lemma 2 again, so by (21) almost no $\alpha$ lies in infinitely many $S_k$. Not reconstructed here.

Dependencies

Dirichlet's theorem on simultaneous approximation; Lemma 2 of the paper (p. 142).

Bears on

No problem page is reached by this result. The Croft question it settles is not an Erdős problem in the corpus; Schmidt credits Erdős with drawing his attention to these problems (p. 138).