# Query-to-Communication Lifting Using Low-Discrepancy Gadgets

## TL;DR

This paper introduces a new lifting theorem that extends the class of gadgets with low discrepancy, including logarithmic-size gadgets, thereby broadening the applicability of query-to-communication complexity reductions.

## Contribution

It proves a lifting theorem for all gadgets with logarithmic length and exponentially-small discrepancy, including randomized cases, significantly expanding previous limitations.

## Key findings

- Lifting theorem now applies to all gadgets with logarithmic length and small discrepancy.
- First randomized lifting theorem for logarithmic-size gadgets.
- Generalizes direct-sum theorems for low-discrepancy functions.

## Abstract

Lifting theorems are theorems that relate the query complexity of a function $f:\{0,1\}^{n}\to\{0,1\}$ to the communication complexity of the composed function $f \circ g^{n}$, for some "gadget" $g:\{0,1\}^{b}\times\{0,1\}^{b}\to\{0,1\}$. Such theorems allow transferring lower bounds from query complexity to the communication complexity, and have seen numerous applications in the recent years. In addition, such theorems can be viewed as a strong generalization of a direct-sum theorem for the gadget $g$. We prove a new lifting theorem that works for all gadgets $g$ that have logarithmic length and exponentially-small discrepancy, for both deterministic and randomized communication complexity. Thus, we significantly increase the range of gadgets for which such lifting theorems hold. Our result has two main motivations: First, allowing a larger variety of gadgets may support more applications. In particular, our work is the first to prove a randomized lifting theorem for logarithmic-size gadgets, thus improving some applications of the theorem. Second, our result can be seen as a strong generalization of a direct-sum theorem for functions with low discrepancy.