Schmidt 1969 disproof conjectures diophantine approximations

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lemma_1: Schmidt's subdivision lemma behind Theorem 1: for N > 1 and epsilon > 0 the unit interval has a subdivision of mesh below epsilon whose half-interval test function is unchanged under multiplication by every integer m from 1 to N, whenever x and mx stay in the unit interval and x avoids a set of measure below epsilon.

question_p138: Schmidt's open question whether a measurable set of positive reals in which no quotient of two distinct elements is an integer must have finite measure, with his note added in proof that Erdos reported Szemeredi can show the measure need not be finite.

theorem_1: Schmidt's disproof of the Davenport-Erdős conjecture: there is a strictly increasing real sequence with consecutive ratios tending to one relative to which the multiples of almost every positive alpha are not uniformly distributed, the test function taking the value 1 on the lower halves of the intervals and -1 elsewhere having averages whose absolute values return to 1.

theorem_2: Schmidt's disproof of Croft's conjecture and its weaker form: however slowly psi(rho) decreases to zero, there is an open set S whose measure in (0, rho) is at least rho psi(rho) yet contains only finitely many of alpha, 2 alpha, 3 alpha, ... for almost all alpha > 0, and a measurable S* with the same bound containing only finitely many multiples of every alpha.

theorem_3: Schmidt's converse to Theorem 2: a set S of positive upper density, limsup of mu_S(rho)/rho equal to some c > 0, contains infinitely many multiples m alpha of almost every alpha, so the decay mu_S(rho) = o(rho) of the sets in Theorem 2 is necessary.

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Schmidt, W. M., Disproof of some conjectures on Diophantine approximations. Studia Sci. Math. Hungar. 4 (1969), 137-144 (received April 2, 1968). The site's key Sc69 for Problems 492 and 1195.

The copy read for this card is the REAL-J repository's scan of the whole volume, Studia Scientiarum Mathematicarum Hungarica, Tomus IV, Fasc. 1--4 (1969), 488 physical pages whose OCR text layer garbles the formulas; the paper occupies printed pp. 137--144, which are physical pp. 139--146 (printed p. $n$ is physical p. $n+2$ for this paper; the mapping was confirmed on the page images). The paper's own running header on its first page reads "Studia Scientiarum Mathematicarum Hungarica 3 (1968) 137--144", a misprint: the volume's cover and title page and the page footers give Tomus IV (1969), the citation the site uses. No other page of the volume is cited here. Every statement below was read on rendered page images. No notice is printed in the volume scan (the cover, PDF p. 1, the imprint page, PDF p. 2, which names Akadémiai Kiadó as publisher, and the last page, PDF p. 488, carry no copyright line); the repository's record for the sibling volume of the same journal (https://real-j.mtak.hu/5461/, read 2026-10-02) states no copyright, license or terms, and this volume's own record (https://real-j.mtak.hu/5455/) was not read for terms; the term is unstated.

Read status: claims checked for the definitions (1)--(5), the account of the earlier positive results and Theorem 1 (printed p. 137, physical p. 139), read clause by clause on the page image for Problem 492; the statements of Theorems 2 and 3, the remarks and the question with its footnote 2 (all on p. 138) and Lemma 1 (pp. 138--139) read clause by clause on the page images; the proofs of Lemma 1 and Theorem 1 (pp. 139--141), of Theorem 2 with Lemma 2 (Section 4, pp. 141--143) and of Theorem 3 with Lemma 3 (Section 5, pp. 143--144) read for their structure and not checked. Nothing here is independently reviewed.

Schmidt disproves two conjectures on Diophantine approximation. Given a strictly increasing sequence x_0 = 0 < x_1 < x_2 < ... with x_n tending to infinity and x_{n+1}/x_n tending to 1, Davenport and Erdos had conjectured that alpha, 2 alpha, 3 alpha, ... is uniformly distributed relative to x_n for almost all alpha > 0; Theorem 1 refutes this by constructing a function f of the associated type with limsup_N |N^{-1} sum_{n <= N} f(alpha n)| = 1 for almost every alpha > 0. Theorem 2 refutes a conjecture of H. T. Croft that any set S of positive reals of infinite Lebesgue measure contains infinitely many multiples of almost every alpha, and even its weaker form asking for one such alpha: for any psi(rho) with 0 < psi(rho) < 1 decreasing to zero there is an open set S whose measure inside (0, rho) satisfies mu_S(rho) >= rho psi(rho) yet for almost all alpha > 0 only finitely many of alpha, 2 alpha, 3 alpha, ... lie in S, and a measurable S* with the same bound that contains only finitely many multiples of every alpha. Theorem 3 shows this decay condition is necessary: when limsup_{rho -> infinity} mu_S(rho)/rho = c > 0 (positive upper density), almost every alpha has infinitely many multiples m alpha in S. The construction for Theorem 1 rests on a subdivision lemma (Lemma 1) proved with Dirichlet's theorem on simultaneous approximation, which is used to build a function f* that is invariant under multiplication by each integer m <= N outside an exceptional set of small measure; Theorem 2's construction uses Dirichlet's theorem again, with a measure estimate (Lemma 2). Schmidt credits Erdos with drawing his attention to these problems, leaves open whether a measurable set S in which no ratio of two distinct elements is an integer must have finite measure, and records in a note added in proof that Erdos informed him Szemeredi can show mu(S) need not be finite.

Source: https://real-j.mtak.hu/5455/.

Bears on. #492: Theorem 1 (printed p. 137, physical p. 139, page image) is the site's "The general conjecture is false, as shown by Schmidt [Sc69]": a real sequence $x_0=0<x_1<\cdots$ with $x_{n+1}/x_n\to1$ relative to which the multiples of almost every $\alpha>0$ are not uniformly distributed, the half-interval test function (5) having $\limsup|N^{-1}\sum_{n\le N}f(\alpha n)|=1$; p. 137 also attests the positive cases (LeVeque and Davenport--LeVeque for monotonic $x_{n+1}-x_n$; Davenport--Erdős for $x_n\gg n^{1/2+\delta}$) and names Davenport and Erdős as the conjecture's authors; the constructed sequence has gaps tending to zero (p. 141), so it is not a set of integers (theorem_1; its subdivision lemma is lemma_1). #1195: the question Schmidt leaves open on printed p. 138 (physical p. 140), whether a measurable set of positive reals in which the quotient of two distinct elements is never an integer must have finite measure, and its footnote 2, added in proof, that Erdős told him Szemerédi can show the measure need not be finite (question_p138); the problem asks how fast the measure of such a set can grow, which the paper does not address.

Results.

• Theorem 1 (p. 137): a strictly increasing real sequence $x_0=0<x_1<\cdots$ with $x_n\to\infty$ and $x_{n+1}/x_n\to1$ whose half-interval test function (5) has $\limsup_N|N^{-1}\sum_{n\le N}f(\alpha n)|=1$ for almost every $\alpha>0$, disproving the Davenport--Erdős conjecture.
• Theorem 2 (p. 138): for $\psi$ with $0<\psi(\varrho)<1$ decreasing to zero, an open set $S$ with $\mu_S(\varrho)\ge\varrho\psi(\varrho)$ containing only finitely many multiples of almost every $\alpha>0$, and a measurable $S^*$ with the same bound containing only finitely many multiples of every $\alpha$, disproving Croft's conjecture and its weaker form.
• Theorem 3 (p. 138): if $\limsup_{\varrho\to\infty}\mu_S(\varrho)/\varrho=c>0$, almost every $\alpha$ has infinitely many multiples $m\alpha$ in $S$.
• Lemma 1 (pp. 138--139): for $N>1$ and $\varepsilon>0$, a subdivision of the unit interval of mesh below $\varepsilon$ whose test function satisfies $f(x)=f(mx)$ for every integer $1\le m\le N$ whenever $x,mx$ lie in the unit interval and $x$ lies outside a set $\sigma_\varepsilon$ of measure less than $\varepsilon$.
• Question (p. 138, with footnote 2): must a measurable set of positive reals in which no quotient of two distinct elements is an integer have finite measure; added in proof, Erdős reported that Szemerédi can show it need not.

Lemmas 2 and 3 (pp. 142, 143) serve only the proofs of Theorems 2 and 3 and are described on those pages.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.