Chen 2005 disjoint arithmetic progressions

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lemma_1: States the exact fixed-parameter smooth-number estimate imported from Canfield, Erdős and Pomerance.

lemma_2: Records the imported prime-factor tail and links its complete canonical proof.

lemma_3: Proves the large-Omega estimate when every prime exponent is bounded by a fixed integer.

lemma_4: Proves the high-prime-power tail with its explicit factor three and real cutoff endpoints.

lemma_5: Proves the common-prime and common-residue pigeonholes without a squarefree hypothesis.

lemma_6: Proves the half-constant disjoint-progression bound for any fixed bound on prime exponents.

theorem: Removes the bounded-exponent hypothesis using high-power counting, residue selection and uniform rescaling.

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Yong-Gao Chen, On disjoint arithmetic progressions, Acta Arithmetica 118.2 (2005), 143–148. DOI and publisher record.

Source and scope

The canonical six-page published PDF is retained byte-for-byte. Its physical pages 1–6 are printed pp. 143–148. The last page records receipt on 23 February 2004 and revision on 13 February 2005. All six pages were visually read; no OCR was used for this reconstruction. The file's text layer carries no copyright or license line; the publisher's issue listing offers the article "Free download under CC-BY license", naming no version or license URL (https://www.impan.pl/en/publishing-house/journals-and-series/acta-arithmetica/all/118/2, read 2026-10-02); the article's own page was not opened.

For distinct moduli $2\le m_1<\cdots<m_s\le x$, let $f(x)$ be the maximum number of pairwise disjoint residue classes. Write $L(c,x)=\exp(c\sqrt{\log x\log\log x})$. Chen proves $f(x)\le x/L(1/2-o(1),x)$ for arbitrary moduli. The precise fixed-loss statement and full proof are in the main theorem.

This extends Croot’s squarefree half-constant bound to all moduli, improving Croot’s unrestricted coefficient $1/6$. It is a historical step toward Problem 202, not a current-best claim. Later sharp results and their status evidence are separate sources.

Complete proof map

The five additional proof components are reconstructed in full.

• Lemma 3 bounds integers with many prime factors counted with multiplicity, when every prime exponent has a fixed upper bound. The multinomial weight and exponential-series tail are proved explicitly.
• Lemma 4 counts large high-exponent parts. Its prime-exponent representation and reciprocal tail include all real cutoff endpoints.
• Lemma 5 extends a common modulus by one prime while retaining a controlled subfamily and residue. It allows repeated prime factors.
• Lemma 6 iterates the selection to obtain the half-constant upper bound for bounded exponents. The common product, residue, surviving family, first-large-prime stop and multiplicity-controlled termination are all explicit.
• The main theorem removes that exponent restriction by two pigeonholes, Chinese remainder compatibility and a uniform change of scale.

The same-paper proof steps are complete. The imported Lemma 1 smooth-number theorem has its exact analytic statement and canonical Canfield–Erdős–Pomerance source link; its original analytic proof remains external. Lemma 2 links the already complete Croot proof, which is not duplicated. Lemma 3 likewise reuses the complete elementary prime-reciprocal estimate.

Reconstruction details

The source’s abbreviated series tail is expanded into a geometric-ratio bound and an integral factorial estimate. Lemma 4’s strict counting comparison is written weakly to include the endpoint one, and its Stieltjes-integral step is replaced by an equivalent nonnegative integral with an explicit atom convention.

Lemma 6 is presented with a stop at the first large prime, avoiding any implicit choice of a terminal chain. The argument relies on $\Omega$, not just $\omega$, because the same prime can be chosen repeatedly. The final proof supplies the coprimality needed to drop the common high-exponent part, the possible quotient modulus one, and uniformity of the scale $x/a$. These are stated expansions of the original argument, not author-issued errata or claims of a new result.

The older Erdős–Szemerédi bounds in the introduction are historical context. Their separate proof is not reconstructed in this source unit. No current-status certification, formal proof or local Lean build is asserted here.

Bears on. Problem 202, through a historical upper bound for disjoint progressions. The common-prime selection also explains the bridge from Croot’s squarefree argument to later methods that retain exact prime-power blocks.