A Sieve Method for Shifted Convolution Sums

TL;DR

This paper introduces an upper-bound sieve method to analyze shifted convolution sums related to Quantum Unique Ergodicity, showing that average cancellation may not be necessary for certain applications.

Contribution

The paper develops a new sieve technique for shifted convolution sums, providing unconditional bounds and broad applicability to similar problems in number theory.

Findings

Unconditional bounds for shifted convolution sums.

Sieve method suggests average size suffices for QUE applications.

Applicable to other multiplicative functions under conditions.

Abstract

We study the average size of shifted convolution summation terms related to the problem of Quantum Unique Ergodicity on ${\rm SL}_2 (\mathbbm{Z})\backslash \mathbbm{H}$. Establishing an upper-bound sieve method for handling such sums, we achieve an unconditional result which suggests that the average size of the summation terms should be sufficient in application to Quantum Unique Ergodicity. In other words, cancellations among the summation terms, although welcomed, may not be required. Furthermore, the sieve method may be applied to shifted sums of other multiplicative functions with similar results under suitable conditions.