Pith. sign in

REVIEW 4 cited by

Partition regularity of generalized Pythagorean pairs

Not yet reviewed by Pith; the record is open.

Re-run · record.json Download PDF Read on arXiv ↗

This paper has not been read by Pith yet. Machine review is queued; the pith claim, tier, and objections will appear here once it completes.

SPECIMEN: schema-true, not a live event

T0 review · schema-true

One-sentence machine reading of the paper's core claim.

pith:XXXXXXXX · record.json · timestamp

arxiv 2407.08360 v3 pith:VPFQZEBS submitted 2024-07-11 math.CO math.NT

Nikos Frantzikinakis , Oleksiy Klurman , Joel Moreira This is my paper · ORCID
classification math.COmath.NT
keywords functionsmultiplicativepartitionpythagoreanquadraticregularityresultsaddress
verification ladder T0 review T1 audit T2 compute T3 formal
0 comments
abstract

We address partition regularity problems for homogeneous quadratic equations. A consequence of our main results is that, under natural conditions on the coefficients $a,b,c$, for any finite coloring of the positive integers, there exists a solution to $ax^2+by^2=cz^2$ where $x$ and $y$ have the same color (and similar results for $x,z$ and $y,z$). For certain choices of $(a,b,c)$, our result is conditional on an Elliott-type conjecture. Our proofs build on and extend previous arguments of the authors dealing with the Pythagorean equation. We make use of new uniformity properties of aperiodic multiplicative functions and concentration estimates for multiplicative functions along arbitrary binary quadratic forms.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Undecidability in the Ramsey theory of polynomial equations and Hilbert's tenth problem

    math.LO 2024-12 accept novelty 8.0 of 10

    Partition regularity of polynomial equations over Z is undecidable if Hilbert's tenth problem over Q is undecidable, and over function fields it is unconditionally Pi_2^0-complete.

  2. On multiplicative recurrence along linear patterns

    math.NT 2024-12 accept novelty 8.0 of 10

    For gcd(a,b,c,d)=1, the liminf of |f(an+b)-f(cn+d)| is zero for every completely multiplicative unit-circle f if and only if a=c and either b=d or a divides bd.

  3. Homogeneous Patterns in Ramsey Theory

    math.CO 2025-01 conditional novelty 6.0 of 10

    The paper establishes new partition regularity results for nonlinear equations, including m-degree homogeneous equations with prescribed degree of regularity.

  4. Partition regularity of homogeneous quadratics: Current trends and challenges

    math.CO 2024-11 accept novelty 1.0 of 10

    A survey of methods proving partition regularity for pairs of variables in homogeneous quadratic equations, with the remaining cases reduced to an open conjecture about vanishing correlations.

Pith tools