Tight universality of $m$-gonal forms with minimal criterion sets

Abstract

For integers $m\geq3$ and $n\geq1$, an $m$-gonal form is called tight $\mathcal{T}(n)$-universal if it represents exactly the positive integers $\mathcal{T}(n)=\{ n, n+1, n+2, \ldots \}$. In this paper, we study the minimal criterion set $\mathrm{CS}(m,n)$ for tight $\mathcal{T}(n)$-universality. Our main result determines $\mathrm{CS}(m,n)$ for $n \geq8$, $3 \leq m \leq \left\lfloor \frac{3n+1}{2} \right\rfloor$, except for $(m,n)=(7,9)$ and $(7,10)$. More precisely, $$ \mathrm{CS}(m,n)= \begin{cases} \{ n, n+1, \ldots, 2n-1 \}, & m=5, \newline \{ n, n+1, \ldots, 2n \}, & m\neq5. \end{cases} $$ We also establish the corresponding tight $\mathcal{T}(n)$-universality results and show that the upper bound on $m$ is optimal.