Kesten 1966 bounded remainder

../

evidence/: Source-owned records of the anchored necessity review and exact historical subjects.

theorem_4: Kesten Theorem 4: bounded discrepancy depends on interval length; exact domains and transfers.

---

Source and identity

Harry Kesten, On a conjecture of Erdős and Szüsz related to uniform distribution mod 1, Acta Arithmetica 12 (1966), 193–212. Journal record. The selected journal header reads XII (1966); the “1966/67” form found in some citations is a bibliographic variant.

The complete digitized article is kesten_1966_bounded_remainder.pdf. The PDF has 11 sheets with two printed pages per sheet; the target article starts on the right-hand leaf of sheet 1, printed p. 193. Each sheet carries the digitizer's "icm©" mark but no license notice; the journal's article page offers the PDF as "Free download under CC-BY license", naming the Creative Commons Attribution license without a version or URL (https://www.impan.pl/en/publishing-house/journals-and-series/acta-arithmetica/all/12/2/96036/on-a-conjecture-of-erdos-and-szusz-related-to-uniform-distribution-mod-1, read 2026-10-02), while its site footer "Copyright © 2026 by IMPAN. All rights reserved." speaks for the site, not the article.

Bounded discrepancy

theorem_4 records the exact length criterion from Theorem 4 on p. 193. For fixed $\xi\in[0,1]$ and a proper interval $0\le a<b\le1$, $b-a<1$, the discrepancy is bounded if and only if $b-a=\{j\xi\}$ for some integer $j$. The theorem constrains the length, and applies to arbitrary admissible translates. It does not force both endpoints individually into the rotation orbit.

This distinction matters for #998. The endpoint converse is explicitly printed in conjecture_p62 and is false, as shown by Alexeev's Lean development. The arbitrary-translate sufficiency estimate has a complete short source proof in equation_3.

Preserved broader coverage and remaining proof compilation

Kesten's printed necessity proof (Section 4, pp. 204–212) is a continued-fraction analysis through the Ostrowski expansion of $N$ in the denominators $q_i$ and the quantities $c_i,q_m,q'_{m+1},a'_{n+1}$. The Theorem 4 page contains an author-recorded reconstruction of the irrational anchored necessity direction: bounded discrepancy for $[0,b)$, $0<b<1$, implies $b=\{j\xi\}$ for an integer $j$. It proves the required continued-fraction identities, partition geometry, both parities, block accumulation and final orbit-point consequence. The proof uses selected multiples of convergent denominators and does not need a general Ostrowski expansion theorem.

Only this anchored irrational necessity reconstruction has independently reviewed proof coverage: the historical whole-claim review returned refutation-failed, and the distinct grade passed the report contract and independence, with documentary corrections. The source-reading record pins the exact reviewed bytes and maps the later prose and standing edits; those edits are not a fresh mathematical review.

The proof consumes printed pp. 193–194, the needed Theorem 1 geometry on pp. 196–199, and Section 4 on pp. 204–212. The source's (4.27)–(4.31) and cases (i)/(ii)/(iii) are bypassed by the local direct exclusion, not reconstructed. The arbitrary-translate reduction cited to Bohl on p. 205 and the rational case remain outside the accepted scope. The earlier accepted Ostrowski sufficiency proof is not re-reviewed, and the endpoint disproof does not depend on this necessity reconstruction. No full local proof coverage of Theorem 4, native claim tier or new problem-status conclusion is asserted. The transformation and source-delta review of this documentary filing was completed and accepted before it was filed.

Beyond Theorem 4 the article also contains the following results. These statements remain a transcription queue; this consolidation does not claim a fresh proof audit of them.

• Theorem 1 and Corollary 1 describe lengths and relative positions of the intervals cut out by the points $\{k\xi\}$, in terms of continued-fraction quantities. Only the geometry consumed by the anchored proof is reconstructed above; the remaining general-$N$ statements and Corollary 1's three-distance conclusion associated with Steinhaus remain outside it.
• Theorem 2 relates the continued-fraction denominators and approximation error to successive fractions in the Farey series $F_N$.
• Theorem 3 gives a metric result for the maximal spacing $L_N(\xi)$ between adjacent rotation points.

Canonical source

This is the canonical home for the complete article identified above. The journal record and local PDF identify the source without a separate acquisition record.

The primary subject is discrepancy, matching Theorem 4's bounded-remainder criterion. Reciprocal problem links support the generated irrationality cross-reference. The consolidation preserves the complete source bytes, correct mathematical content, and historical citation variants while correcting the old endpoint assertion.