Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0706.2782

Extremal subsets of {1,...,n} avoiding solutions to linear equations in three variables

Peter Hegarty
View original source ↗Read source document ↗Export metadata ↓

Abstract

We refine previous results to provide examples, and in some cases precise classifications, of extremal subsets of {1,...,n} containing no solutions to a wide class of non-invariant, homogeneous linear equations in three variables, i.e.: equations of the form ax+by=cz with a+b \neq c.

Explore related subjects

math.NTmath.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Peter Hegarty. 2007-06-19. Extremal subsets of {1,...,n} avoiding solutions to linear equations in three variables. https://arxiv.org/abs/0706.2782

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Co-rank $1$ Arithmetic Siegel--Weil III: Geometric local-to-global

This is the third in a sequence of four papers, where we prove the arithmetic Siegel--Weil formula in co-rank $1$ for Kudla--Rapoport special cycles on exotic smooth integral models of unitary Shimura varieties of arbitrarily large even arithmetic dimension. Our arithmetic Siegel--Weil formula implies that degrees of Kudla--Rapoport arithmetic special $1$-cycles are encoded in the first derivatives of unitary Eisenstein series Fourier coefficients. In this paper, we make the reduction from global arithmetic intersection numbers for special cycles to the local geometric quantities in our companion papers. Building on our previous companion papers, we also propose a construction for arithmetic special cycle classes associated to possibly singular matrices of arbitrary co-rank.

math.NT↗
arXiv2026

Co-rank $1$ Arithmetic Siegel--Weil IV: Analytic local-to-global

This is the fourth in a sequence of four papers, where we prove the arithmetic Siegel--Weil formula in co-rank $1$ for Kudla--Rapoport special cycles on exotic smooth integral models of unitary Shimura varieties of arbitrarily large even arithmetic dimension. Our arithmetic Siegel--Weil formula implies that degrees of Kudla--Rapoport arithmetic special $1$-cycles are encoded in the first derivatives of unitary Eisenstein series Fourier coefficients. In this paper, we pin down precise normalizations for some $U(m,m)$ Siegel Eisenstein series, give local Siegel--Weil special value formulas with explicit constants, and record a geometric Siegel--Weil result for degrees of complex $0$-cycles. Using this, we complete the proof of our arithmetic Siegel--Weil results by patching together the local main theorems from our companion papers.

math.NT↗
arXiv2026

Sub-Weyl bound for $GL(2)$ via trivial delta

For a $SL(2,\mathbb{Z})$ form $f$, we obtain the sub-Weyl bound \begin{equation*} L(1/2+it,f)\ll_{f,\varepsilon} t^{1/3-δ+\varepsilon}, \end{equation*} where $δ=1/174$, thereby crossing the Weyl barrier for the first time beyond $GL(1)$. The proof uses a refinement of the `trivial' delta method.

math.NT↗