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We establish a necessary and sufficient condition for pairs of integers toarise as the weights at the fixed points of an effective circle action on acompact almost complex 4-manifold with a discrete fixed point set. As anapplication, we provide a necessary and sufficient condition for a pair ofintegers to arise as the Chern numbers of such an action, answering negativelya question by Sabatini whether $c_1^2[M] \leq 3 c_2[M]$ holds for any suchmanifold $M$. We achieve this by demonstrating that pairs of integers thatarise as weights of a circle action, also arise as weights of a restriction ofa $\mathbb{T}^2$-action. Furthermore, we discuss applications to circle actionson complex/symplectic 4-manifolds and semi-free circle actions with discretefixed point sets.

Circle actions on four dimensional almost complex manifolds with discrete fixed point sets