Ostrowski 1927 mathematische miszellen

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equation_3: Ostrowski equation (3), with its complete arbitrary-translate fractional-part proof.

evidence/: Mathematical evidence for Ostrowski's translated-interval theorem.

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Ostrowski, Alexander, Mathematische Miszellen. IX. Notiz zur Theorie der Diophantischen Approximationen. Jber. Deutsch. Math.-Verein. 36 (1927), 178-180.

This short German note (read as page images; the scan is legible) concerns the discrepancy of the sequence R(n alpha), the fractional parts of n alpha, for irrational alpha. Writing N(J,x) for the number of R(n alpha), n <= x, lying in a subinterval J, equidistribution gives N(J,x) = |J| x + o(x); Hecke had proved the stronger bounded-remainder statement N(J,x) = |J| x + rho with rho bounded for all x > 0 whenever alpha is irrational and J = [0, R(nu alpha)) for a positive integer nu (display (2), p. 179, citing Hamb. Math. Abh. 1 (1922), 73-74). Ostrowski's theorem strengthens this to arbitrarily placed intervals: for any real alpha, any nonzero integer nu, and any subinterval J of [0,1) of length R(nu alpha), counted with its initial point and without its final point, one has |N(J,x) - |J| x| < |nu| for all integers x > 0 (display (3), p. 179), so J may be translated freely (and may wrap around 0 modulo 1). After reducing to positive nu by complements, the proof introduces a shift parameter xi into the auxiliary sum S_xi(x) = sum_{n=1}^{x+nu} R(n alpha + xi) - sum_{n=1}^{x} R(n alpha + xi) - sum_{n=1}^{nu} R(n alpha + xi), shows the identity S_xi(x) = R(nu alpha) x - N(J,x) (display (4), p. 180), and rewrites the same sum as nu differences of fractional parts, each of absolute value below 1; that shift is the new element compared with Hecke's proof. The note is the source for the bounded-remainder-interval fact relevant to problem 998 on the discrepancy of {n alpha}.

Source: https://gdz.sub.uni-goettingen.de/id/PPN37721857X_0036. The copy read for this card is the four-page scan from that record (424,499 bytes). Its first page is the digitizing library's cover sheet, which states that "Some of our collections are protected by copyright", that "Publication and/or broadcast in any form (including electronic) requires prior written permission" and that reproductions of its material may not be further "reproduced without written permission from the Goettingen State- and University Library"; the article pages (2--4) are image-only and print no notice, every other right reserved.

Bears on. #998: display (3) is the direction of the length criterion opposite to the one the corrected statement asks for: an interval of length $\{j\alpha\}$, $j\ne0$, has discrepancy below $|j|$ at every index, wherever it is placed. The paper does not discuss the converse.

Results.

• Equation (3) (p. 179): for real alpha, nonzero integer nu and any subinterval J of [0,1) of length R(nu alpha), translated freely modulo 1, |N(J,x) - |J|x| < |nu| for all integers x > 0; the page also restates the identity (4) (p. 180) used in its proof.

Extracted proof for #998

equation_3 states equation (3) on printed p. 179 and reconstructs its complete proof through p. 180, including arbitrary real parameters, translated half-open circle intervals, signed nonzero integer multiples, and the negative-index complement argument. The bound is strictly less than the absolute value of the integer multiple.

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