On non-intersecting arithmetic progressions

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corollary_to_theorem_1: Grouping by the powerful part and a common residue reduces arbitrary distinct moduli to the squarefree theorem and gives coefficient one sixth.

lemma_1: Croot invokes the Canfield–Erdős–Pomerance smooth-number estimate at y=exp(c sqrt(log x log-log x)); its proof is an external input.

lemma_2: A factorial-moment count bounds integers with at least c sqrt(log x over log-log x) distinct prime factors.

lower_bound: A Chinese-remainder construction and the smooth-number input give f(x) at least x times exp(-(sqrt(2)+o(1))sqrt(log x log-log x)).

powerful_tail: Powerful integers have an O(sqrt(t)) counting bound and a uniform weighted tail, controlling the nonsquarefree parts of the moduli.

prime_reciprocal_bound: An elementary zeta comparison bounds the reciprocal-prime sum and the finite Euler product needed for the smooth prime-power estimate.

selection_lemma: Disjoint squarefree congruences with few prime factors admit a nested residue selection that terminates at a new large common divisor.

smooth_prime_powers: A weighted prime-power union bound proves that prime-power smoothness has the same main logarithmic count as ordinary smoothness.

theorem_1: A disjoint family with distinct squarefree moduli at most x has size at most x times exp(-(1/2-o(1))sqrt(log x log-log x)).

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Ernest S. Croot III, “On non-intersecting arithmetic progressions,” Acta Arithmetica 110 (2003), no. 3, 233–238. DOI: 10.4064/aa110-3-3.

Source versions

The canonical published PDF is the six-page journal version, acquired from the publisher on 5 September 2026. The publisher lists a free download under CC-BY. The published PDF prints no copyright or license line; the publisher's article page offers the PDF "Free download under CC-BY license", naming no version (https://www.impan.pl/en/publishing-house/journals-and-series/acta-arithmetica/all/110/3/82874/on-non-intersecting-arithmetic-progressions, read 2026-10-02), and its footer "Copyright © 2026 by IMPAN. All rights reserved." speaks for the site, not the paper. For the arXiv v1 PDF, the arXiv record carries no license field, so arXiv's assumed license applies (arXiv:math/0208236), every other right reserved. The author manuscript PDF prints no copyright or license line, and the author's home page that linked it states no copyright, license or terms and no longer lists the file (https://ecroot.math.gatech.edu/, read 2026-10-02); the term is unstated.

The author manuscript read for this card is the five-page version available from the author's site. The arXiv version read is the five-page math/0208236v1, dated 30 August 2002.

All pages of the three versions were read for this compilation. The numbered results and mathematical proof route agree. The journal rearranges the text across six pages, omits the manuscript abstract, adds publication information, and corrects the Canfield–Erdős–Pomerance reference year from 1980 to 1983. The definition and proof issues identified below persist in the published version. All result-page locators refer to journal pages.

Results and proof coverage

Let $f(x)$ be the maximum number of pairwise disjoint integer congruence classes with distinct moduli in $[2,x]$, and put $T(x)=\sqrt{\log x\log\log x}$. The paper proves that, for each $\eta>0$ and sufficiently large $x$,

$$x e^{-(\sqrt2+\eta)T(x)} \le f(x)\le x e^{-(1/6-\eta)T(x)}.$$

For squarefree moduli the upper coefficient improves to $1/2$. These are historical bounds for Problem 202, not a claim of the current sharp answer.

The lower-bound construction orders prime-power factors and prescribes a chain of CRT residues. The first point where two chains differ proves disjointness. Its count uses prime-power smoothness, deduced from the external smooth-number estimate in Lemma 1.

The squarefree upper bound discards integers with too many distinct prime factors using Lemma 2, then discards smooth moduli. A nested residue-selection argument forces a large prime divisor into many of the remaining moduli. The general-modulus corollary groups by a common powerful part and a common residue, then applies the squarefree theorem on the correct smaller scale.

Lemma 1 is an exact external theorem statement, not a reconstructed proof of its Canfield–Erdős–Pomerance source.

Corrections and qualifications

• The smooth-count definitions print $n\le y$ instead of $n\le x$.
• Equation (2) uses a square-divisor comparison that does not capture every excluded prime power.
• In the construction, the prime-power factors less than $p$ are those of $q/p$; the literal condition on all factors of a multiple of $p$ is empty.
• The residue-selection proof does not show that its terminal prime is new, as property 4 requires.

No Lean build, later sharp bound, or current-status determination is supplied by this source unit.

Bears on. #202: the construction on p. 234 and the Corollary to Theorem 1 bound the problem's maximum $f(N)$ below and above on the scale $\sqrt{\log N\log\log N}$, with coefficients $\sqrt2$ and $1/6$; Theorem 1 gives the upper coefficient $1/2$ only for families whose moduli are all squarefree. The paper does not determine the sharp coefficient.

Only the edition under an open license is held; the source's other editions are not, since no license on record permits their redistribution, and the card cites the edition it names above.