Explicit small sets with ε-discrepancy on Bohr sets

• Bohr sets (BS) are a higher rank analogue of arithmetic progressions (AP).
• Razborov–Szemeredi–Wigderson: ∃ explicit set with ε  -discrepancy on all AP in ZNZN.
• This work: ∃ explicit set with ε-discrepancy on all rank d   BS in ZNZN.
• Proof method: reduction to the existence of explicit ε′ε′-biased sets in ZNZN.

We prove that there are explicit sets A that have ε-discrepancy on all rank d   Bohr sets in ZNZN and are of size |A|=poly((lnN)d,1/ε). This extends the result on explicit sets with ε-discrepancy on arithmetic progressions Razborov et al. [2] (1993) to Bohr sets which are the higher rank analogue of arithmetic progressions. Our proof is via a reduction to the existence of explicit small biased sets in ZNZN.