On two non-existence results for Cameron-Liebler k-sets in PG(n,q)

This paper focuses on non-existence results for Cameron-Liebler k-sets. A Cameron-Liebler k-set is a collection of k-spaces in PG(n,q) or AG(n,q) admitting a certain parameter x, which is dependent on the size of this collection. One of the main research questions remains the (non-)existence of Cameron-Liebler k-sets with parameter x. This paper improves two non-existence results. First we show that the parameter of a non-trivial Cameron-Liebler k-set in PG(n,q) should be larger than qn-5k/2-1, which is an improvement of an earlier known lower bound. Secondly, we prove a modular equality on the parameter x of Cameron-Liebler k-sets in PG(n,q) with x<qⁿ⁻ᵏ-1qᵏ⁺¹-1, n≥ 2k+1, n-k+1≥ 7 and $n-k$ even. In the affine case we show a similar result for n-k+1≥ 3 and $n-k$ even. This is a generalization of earlier known modular equalities in the projective and affine case.