Maximum-load bounds for almost-linear integer hashing

Determine whether, for almost-linear integer hash functions of the form h_{a,b}(x) = floor(((a x + b) mod r)/u) mapping N elements into a range of size N, one can guarantee a maximum bucket load strictly better than N^{1/3}.

Background

The paper’s reduction for k-SUM relies on hashing schemes over integers. Known bounds for the maximum load of these almost-linear functions (e.g., Dietzfelbinger-style hashing) are insufficiently strong for directly replicating certain linear-hashing properties available in the XOR setting.

The authors note that improving these bounds is a recognized challenge in hashing theory and would simplify or strengthen reductions of the type developed in the paper.