Local distribution of arithmetic dilates
Lead Research Organisation:
KING'S COLLEGE LONDON
Abstract
Analytic number theorists study how arithmetic sequences are modelled by random processes. This project aims to show that a property of the square numbers (involving gaps between dilated squares) is approximated by a Poisson process. Doing so would resolve a deep 25-year-old conjecture, and provide rigorous justification around the mysterious phenomenon of quantum chaos.
Context
Start with an irrational number x, e.g. the square root of 2, and consider x, 2x, 3x,... etcetera. Restricting to the part after the decimal point (i.e. working ‘mod 1’), the sequence becomes 0.414...,0.828...,0.242...: what is its behaviour? In 1916 Weyl proved that nx mod 1 is equidistributed, meaning that the proportion of numbers n for which nx mod 1 lies in a fixed interval is approximately the length of that interval. For example, nx mod 1 begins 0.375569... for about 1 in a million values of n.
A sequence of N independent uniformly distributed random points between 0 and 1 is also equidistributed (with high probability). Thus, Weyl demonstrated a pseudorandomness property of the sequence nx mod 1; he also proved the same result with n replaced by the sequence of square numbers. These results have become central number-theoretic tools, as (when combined with analytic techniques) they help to count solutions to equations.
Consider again the N random points. The average gap between consecutive points is 1/N, albeit with some much tighter clusters. These clusters can be precisely calculated: for instance, the proportion of gaps of length at most s/N is about 1 - exp(-s). Now, take the first N squares 1,4,9,16... and consider the sequence x,4x,9x,16x, etcetera. The sequence mod 1 runs 0.414...,0.656...,0.727...,0.627....Weyl showed the sequence is pseudorandom, in the sense of equidistribution. But does its gap distribution agree with the random points?
Rudnick-Sarnak conjectured that the distributions do agree, at least for a generic x. This is a profound and influential observation, positing a vast generalisation of Weyl's work.
Aims, challenges, and objectives
The ultimate goal of this project is to prove the Rudnick-Sarnak conjecture. This would be a major result in arithmetic pseudorandomness. There is a natural approach, using so-called 'k-correlation functions', and in earlier work I advanced the study of 2-correlations and 3-correlations associated to this problem. The project's sub-goals build on this work, before generalising to all k-correlations.
The Rudnick-Sarnak prediction is backed by convincing data. However, little has been rigorously proved, the main challenge being that the scale length 1/N is so short.
Potential applications and benefits
The main beneficiaries will be academic, as the topic intersects with many mathematical areas: number theory, harmonic analysis, probability, additive combinatorics, and dynamics.
There is also a potential application to physics. The sequence of dilated squares can be viewed as the eigenvalue energy levels of a quantum system; Berry-Tabor observed in experiments that the gap distributions of such eigenvalues are determined by the dynamics of the corresponding classical system. Integrable classical dynamics should correspond to a 'poissonian gap distribution' (our case): chaotic dynamics should correspond to gap distributions arising from random matrix theory. Though there is strong empirical evidence, precious little has been rigorously proved: this project aims to provide such proof in an explicit case.
As this project studies pseudorandom properties of concrete sequences, there could also be potential applications to Monte Carlo methods: in mathematical finance, engineering, and beyond.
Context
Start with an irrational number x, e.g. the square root of 2, and consider x, 2x, 3x,... etcetera. Restricting to the part after the decimal point (i.e. working ‘mod 1’), the sequence becomes 0.414...,0.828...,0.242...: what is its behaviour? In 1916 Weyl proved that nx mod 1 is equidistributed, meaning that the proportion of numbers n for which nx mod 1 lies in a fixed interval is approximately the length of that interval. For example, nx mod 1 begins 0.375569... for about 1 in a million values of n.
A sequence of N independent uniformly distributed random points between 0 and 1 is also equidistributed (with high probability). Thus, Weyl demonstrated a pseudorandomness property of the sequence nx mod 1; he also proved the same result with n replaced by the sequence of square numbers. These results have become central number-theoretic tools, as (when combined with analytic techniques) they help to count solutions to equations.
Consider again the N random points. The average gap between consecutive points is 1/N, albeit with some much tighter clusters. These clusters can be precisely calculated: for instance, the proportion of gaps of length at most s/N is about 1 - exp(-s). Now, take the first N squares 1,4,9,16... and consider the sequence x,4x,9x,16x, etcetera. The sequence mod 1 runs 0.414...,0.656...,0.727...,0.627....Weyl showed the sequence is pseudorandom, in the sense of equidistribution. But does its gap distribution agree with the random points?
Rudnick-Sarnak conjectured that the distributions do agree, at least for a generic x. This is a profound and influential observation, positing a vast generalisation of Weyl's work.
Aims, challenges, and objectives
The ultimate goal of this project is to prove the Rudnick-Sarnak conjecture. This would be a major result in arithmetic pseudorandomness. There is a natural approach, using so-called 'k-correlation functions', and in earlier work I advanced the study of 2-correlations and 3-correlations associated to this problem. The project's sub-goals build on this work, before generalising to all k-correlations.
The Rudnick-Sarnak prediction is backed by convincing data. However, little has been rigorously proved, the main challenge being that the scale length 1/N is so short.
Potential applications and benefits
The main beneficiaries will be academic, as the topic intersects with many mathematical areas: number theory, harmonic analysis, probability, additive combinatorics, and dynamics.
There is also a potential application to physics. The sequence of dilated squares can be viewed as the eigenvalue energy levels of a quantum system; Berry-Tabor observed in experiments that the gap distributions of such eigenvalues are determined by the dynamics of the corresponding classical system. Integrable classical dynamics should correspond to a 'poissonian gap distribution' (our case): chaotic dynamics should correspond to gap distributions arising from random matrix theory. Though there is strong empirical evidence, precious little has been rigorously proved: this project aims to provide such proof in an explicit case.
As this project studies pseudorandom properties of concrete sequences, there could also be potential applications to Monte Carlo methods: in mathematical finance, engineering, and beyond.
Organisations
People |
ORCID iD |
| Aled Walker (Principal Investigator) |