Doorn 2025 improved bounds mayer erdos phenomenon similarly

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lemma_2: Van Doorn's 2025 lemma behind his lower bound for Problem 1005: two Farey fractions of order n that enclose a Farey fraction a/b are similarly ordered whenever their index distance is at most (n + b + 1)/(2b).

theorem_1: Van Doorn's 2025 upper bound for the largest guaranteed run of similarly ordered Farey fractions of order n: f(n) is at most floor(n/4) + d with d = 1, 2, 2, 4 according to n modulo 4, by explicit badly ordered pairs around 1/2; conjectured to be exact for all n >= 92 and checked up to 5000.

theorem_2: Van Doorn's 2025 lower bound for Problem 1005: two Farey fractions of order n at index distance at most (n/12)(1 - 4 n^{-1/3}) are similarly ordered, so f(n) is at least (1/12 - o(1)) n; an optimization of Erdős's 1943 argument, improving the constants 1/400 (Erdős) and 1/480 (Zaharescu).

theorem_3: Van Doorn's 2025 local-density form of his lower bound for Problem 1005: two Farey fractions of order n that differ by x/n either enclose a Farey fraction of denominator below 6/x or are more than nx(1/12 - o(1)) places apart in the sequence.

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Wouter van Doorn, Improved bounds for the Mayer-Erdős phenomenon on similarly ordered Farey fractions, arXiv:2509.00121 (2025). The site's key vD25b.

The copy read for this card is arXiv:2509.00121v1 (28 August 2025), 9 pages with a text layer whose plus signs drop out of the extracted text; the statements below were read on the rendered page images of pp. 1--9. The abstract page lists one version and no journal reference, and no journal record was found (Crossref bibliographic query, 2026-09-18): a preprint. Source: https://arxiv.org/abs/2509.00121. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2509.00121), every other right reserved.

Read status: claims checked for the definition of $f(n)$, the history paragraph, Theorem 1 with the Conjecture and the list of exceptional $n$, Lemmas 1--6 as statements, Theorem 2 and Theorem 3, read clause by clause on the page images (pp. 1--8); the proofs of Theorems 1 and 2 were read for structure and not checked.

For the Farey sequence $a_1/b_1,a_2/b_2,\ldots$ of order $n\ge4$, van Doorn studies $f(n)$, the largest integer such that $(a_l-a_k)(b_l-b_k)\ge0$ (the fractions are "similarly ordered") whenever $|l-k|\le f(n)$; the condition $n\ge4$ ensures a non-similarly-ordered pair exists ($1/4$ and $2/3$). Theorem 1 gives the upper bound $f(n)\le\lfloor n/4\rfloor+d$ with $d=1,2,2,4$ according to $n\bmod4$, via explicit Farey neighborhoods of fractions like $(2m-1)/(4m)$ together with the standard criterion $bc-ad=1$ and $\max(b,d)\le n<b+d$ for consecutive Farey fractions; a conjecture (checked for all $n\le5000$, with the fifteen exceptions below $92$ listed) asserts this is exactly $f(n)$ for $n\ge92$. On the lower side, Lemma 2 shows that fractions bracketing a fraction $a/b$ of small denominator are similarly ordered whenever $l-k\le(n+b+1)/(2b)$, using the fact that numerators and denominators of Farey fractions adjacent to $a/b$ form arithmetic progressions; optimizing Erdős's argument, Theorem 2 yields $f(n)\ge\lfloor(n/12)(1-4/n^{1/3})\rfloor=(1/12-o(1))n$, improving Erdős's constant $1/400$ and Zaharescu's $1/480$ (the paper's account: Zaharescu generalized to arbitrary linear forms with $c=1/480$, and Meng and Zaharescu to several variables). The paper reports no earlier improvement on Erdős's constant that its author was aware of; the site's account of Problem 1005 now rests on a 2026 preprint that claims the matching lower bound $(1/4-o(1))n$.

Contents

• Introduction (p. 1): the definition of $f(n)$; Mayer proved $f(n)\ge3$ for $n\ge5$ (his first 1942 paper) and $f(n)\to\infty$ (his second); Erdős [3] proved $f(n)>cn$, "his proof showed that one can take $c=1/400$"; Zaharescu [4] ($c=1/480$, linear forms) and Meng--Zaharescu [5]; "no improvements have occurred in the literature" since.
• Theorem 1 (p. 2): $f(n)\le\lfloor n/4\rfloor+d$, $d=1,2,2,4$ for $n\equiv0,1,2,3 \pmod4$, for all $n\ge4$; Lemma 1 (consecutive Farey fractions); the proof by the explicit segments around $(2m-1)/(4m)$ and $2m/(4m+1)$; the Conjecture ($f(n)>n/4$ for all $n\ge4$; $f(n)=\lfloor n/4\rfloor+d$ for all $n\ge92$), checked for $n\le5000$, with the exceptions $n=7,9,11,15, 19,23,25,27,31,35,39,49,51,63,91$; a stronger classification conjecture.
• Section 3 (pp. 3--8): Lemma 2 (p. 3; bracketing a fraction $a/b$: similarly ordered if $l-k\le(n+b+1)/(2b)$); Lemma 3 ($N>n^2/4$ Farey fractions of order $n$); Lemma 4 (Dress's discrepancy bound $N(\alpha-1/n)\le A_n(\alpha)\le N(\alpha+1/n)$); Theorem 2 (p. 5): if $k<l$ and $l-k\le\frac n{12}(1-\frac4{n^{1/3}})$ then $a_k/b_k$ and $a_l/b_l$ are similarly ordered; Lemmas 5--6 and the proof (pp. 5--8) by splitting $\sum1/(b_ib_{i+1})$ as in Erdős.
• Section 4 (p. 8): Theorem 3, a local-density form (either a Farey fraction with denominator below $6/x$ lies between the two fractions or $l-k>nx(1/12-o(1))$ when they are $x/n$ apart); the remark that for a badly ordered pair with $a_k/b_k\ge1/2-o(1)$ the argument gives $l-k>n(1/8-o(1))$, "at most a factor 2 off from optimal".

Compiled scope

The whole preprint was read. Theorems 1, 2 and 3 and Lemma 2 are compiled as statements with proof pointers; the computation behind the Conjecture was not rerun; no step of the proofs was checked and nothing here is independently reviewed.

Bears on. #1005: the same function $f(n)$ as the problem's; Theorem 1 is the upper bound $f(n)\le\lfloor n/4\rfloor+d$ the site prints as $f(n)\le n/4+O(1)$, and Theorem 2 is the lower bound $(1/12-o(1))n$ the site prints; the Conjecture that the upper bound is exact for $n\ge92$ is the statement the 2026 preprints address (asymptotically, and then exactly). Lemma 2 and Theorem 3 are not bounds on $f(n)$: Lemma 2 gives similar ordering for pairs enclosing a Farey fraction $a/b$ when $l-k\le(n+b+1)/(2b)$ and is a step in the proof of Theorem 2, and Theorem 3 is the general form, in terms of the gap $x/n$, of what that proof gives; neither improves the bound of Theorem 2. Theorems 1 and 2 are bounds on the problem's $f(n)$ that the paper proves; the Conjecture is stated there, not proved.

Results to transcribe.

• Theorem 1: for all $n\ge4$, $f(n)\le\lfloor n/4\rfloor+d$ with $d=1,2,2,4$ depending on $n\bmod4$; in particular some $k<l<k+n/4+5$ give $(a_l-a_k)(b_l-b_k)<0$.
• Theorem 2 (the lower bound): $f(n)\ge\lfloor\frac n{12}(1-4n^{-1/3})\rfloor$, hence $f(n)\ge(1/12-o(1))n$, improving Erdős's constant $1/400$ and Zaharescu's $1/480$.
• Lemma 2 (p. 3): if $a_k/b_k\le a/b\le a_l/b_l$ are Farey fractions of order $n$ and $l-k\le(n+b+1)/(2b)$, then $a_k/b_k$ and $a_l/b_l$ are similarly ordered.
• Theorem 3 (p. 8): if $a_l/b_l-a_k/b_k=x/n$ with $x>0$, then either a Farey fraction $a/b$ with $b<6/x$ satisfies $a_k/b_k\le a/b\le a_l/b_l$, or $l-k>nx(1/12-o(1))$.
• Conjecture: for $n\ge92$, $f(n)$ equals $\lfloor n/4\rfloor+d$ exactly with $d$ as in Theorem 1; checked by computer for all $n\le5000$.

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