On the almost universality of $\lfloor x^2/a\rfloor+\lfloor y^2/b\rfloor+\lfloor z^2/c\rfloor$
Hai-Liang Wu, He-Xia Ni, Hao Pan
TL;DR
The paper advances the study of representing natural numbers by ternary floor-sum forms $\mathcal F_{a,b,c}(x,y,z)=\lfloor x^2/a\rfloor+\lfloor y^2/b\rfloor+\lfloor z^2/c\rfloor$ by proving almost universality results. It shows that for every $m\ge 3$, Farhi's conjecture holds for all sufficiently large $n$, and that if $a,b,c\ge 5$ are pairwise coprime, Sun's conjecture holds for all sufficiently large $n$, i.e., these ternary floor-sum forms are almost universal. The approach translates the problem to shifted-lattice representations $l_{a,b,c}(n)$ and uses congruence theta functions to decompose the associated modular forms into Eisenstein, unary-theta, and cusp components, then exploits local-to-global principles via the Local Square Theorem. The results generalize known cases and provide a robust framework for almost universality of diagonal floor-sum representations.
Abstract
In 2013, Farhi conjectured that for each $m\geq 3$, every natural number $n$ can be represented as $\lfloor x^2/m\rfloor+\lfloor y^2/m\rfloor+\lfloor z^2/m\rfloor$ with $x,y,z\in\Z$, where $\lfloor\cdot\rfloor$ denotes the floor function. Moreover, in 2015, Sun conjectured that every natural number $n$ can be written as $\lfloor x^2/a\rfloor+\lfloor y^2/b\rfloor+\lfloor z^2/c\rfloor$ with $x,y,z\in\Z$, where $a,b,c$ are integers and $(a,b,c)\neq (1,1,1),(2,2,2)$. In this paper, with the help of congruence theta functions, we prove that for each $m\geq 3$, Farhi's conjecture is true for every sufficiently large integer $n$. And for $a,b,c\geq 5$ with $a,b,c$ are pairwisely co-prime, we also confirm Sun's conjecture for every sufficiently large integer $n$.