ECCC-Report TR23-133https://eccc.weizmann.ac.il/report/2023/133Comments and Revisions published for TR23-133en-usTue, 13 Feb 2024 20:51:38 +0200
Revision 2
| Product mixing in compact Lie groups |
David Ellis,
Guy Kindler,
Noam Lifshitz,
Dor Minzer
https://eccc.weizmann.ac.il/report/2023/133#revision2If $G$ is a group, we say a subset $S$ of $G$ is product-free if the equation $xy=z$ has no solutions with $x,y,z \in S$. For $D \in \mathbb{N}$, a group $G$ is said to be $D$-quasirandom if the minimal dimension of a nontrivial complex irreducible representation of $G$ is at least $D$. Gowers showed that in a $D$-quasirandom finite group $G$, the maximal size of a product-free set is at most $|G|/D^{1/3}$. This disproved a longstanding conjecture of Babai and S\'os from 1985.
For the special unitary group, $G=SU(n)$, Gowers observed that his argument yields an upper bound of $n^{-1/3}$ on the measure of a measurable product-free subset. In this paper, we improve Gowers' upper bound to $\exp(-cn^{1/3})$, where $c>0$ is an absolute constant. In fact, we establish something stronger, namely, product-mixing for measurable subsets of $SU(n)$ with measure at least $\exp(-cn^{1/3})$; for this product-mixing result, the $n^{1/3}$ in the exponent is sharp.
Our approach involves introducing novel hypercontractive inequalities, which imply that the non-Abelian Fourier spectrum of the indicator function of a small set concentrates on high-dimensional irreducible representations.
Our hypercontractive inequalities are obtained via methods from representation theory, harmonic analysis, random matrix theory and differential geometry. We generalize our hypercontractive inequalities from $SU(n)$ to an arbitrary $D$-quasirandom compact connected Lie group for $D$ at least an absolute constant, thereby extending our results on product-free sets to such groups.
We also demonstrate various other applications of our inequalities to geometry (viz., non-Abelian Brunn-Minkowski type inequalities), mixing times, and the theory of growth in compact Lie groups. A subsequent work due to Arunachalam, Girish and Lifshitz uses our inequalities to establish new separation results between classical and quantum communication complexity.Tue, 13 Feb 2024 20:51:38 +0200https://eccc.weizmann.ac.il/report/2023/133#revision2
Revision 1
| Product mixing in compact Lie groups |
David Ellis,
Guy Kindler,
Noam Lifshitz,
Dor Minzer
https://eccc.weizmann.ac.il/report/2023/133#revision1If $G$ is a group, we say a subset $S$ of $G$ is product-free if the equation $xy=z$ has no solutions with $x,y,z \in S$. For $D \in \mathbb{N}$, a group $G$ is said to be $D$-quasirandom if the minimal dimension of a nontrivial complex irreducible representation of $G$ is at least $D$. Gowers showed that in a $D$-quasirandom finite group $G$, the maximal size of a product-free set is at most $|G|/D^{1/3}$. This disproved a longstanding conjecture of Babai and S\'os from 1985.
For the special unitary group, $G=SU(n)$, Gowers observed that his argument yields an upper bound of $n^{-1/3}$ on the measure of a measurable product-free subset. In this paper, we improve Gowers' upper bound to $\exp(-cn^{1/3})$, where $c>0$ is an absolute constant. In fact, we establish something stronger, namely, product-mixing for measurable subsets of $SU(n)$ with measure at least $\exp(-cn^{1/3})$; for this product-mixing result, the $n^{1/3}$ in the exponent is sharp.
Our approach involves introducing novel hypercontractive inequalities, which imply that the non-Abelian Fourier spectrum of the indicator function of a small set concentrates on high-dimensional irreducible representations.
Our hypercontractive inequalities are obtained via methods from representation theory, harmonic analysis, random matrix theory and differential geometry. We generalize our hypercontractive inequalities from $SU(n)$ to an arbitrary $D$-quasirandom compact connected Lie group for $D$ at least an absolute constant, thereby extending our results on product-free sets to such groups.
We also demonstrate various other applications of our inequalities to geometry (viz., non-Abelian Brunn-Minkowski type inequalities), mixing times, and the theory of growth in compact Lie groups. A subsequent work due to Arunachalam, Girish and Lifshitz uses our inequalities to establish new separation results between classical and quantum communication complexity.Mon, 02 Oct 2023 12:37:29 +0300https://eccc.weizmann.ac.il/report/2023/133#revision1
Paper TR23-133
| Product mixing in compact Lie groups |
David Ellis,
Guy Kindler,
Noam Lifshitz,
Dor Minzer
https://eccc.weizmann.ac.il/report/2023/133If $G$ is a group, we say a subset $S$ of $G$ is product-free if the equation $xy=z$ has no solutions with $x,y,z \in S$. For $D \in \mathbb{N}$, a group $G$ is said to be $D$-quasirandom if the minimal dimension of a nontrivial complex irreducible representation of $G$ is at least $D$. Gowers showed that in a $D$-quasirandom finite group $G$, the maximal size of a product-free set is at most $|G|/D^{1/3}$. This disproved a longstanding conjecture of Babai and S\'os from 1985.
For the special unitary group, $G=SU(n)$, Gowers observed that his argument yields an upper bound of $n^{-1/3}$ on the measure of a measurable product-free subset. In this paper, we improve Gowers' upper bound to $\exp(-cn^{1/3})$, where $c>0$ is an absolute constant. In fact, we establish something stronger, namely, product-mixing for measurable subsets of $SU(n)$ with measure at least $\exp(-cn^{1/3})$; for this product-mixing result, the $n^{1/3}$ in the exponent is sharp.
Our approach involves introducing novel hypercontractive inequalities, which imply that the non-Abelian Fourier spectrum of the indicator function of a small set concentrates on high-dimensional irreducible representations.
Our hypercontractive inequalities are obtained via methods from representation theory, harmonic analysis, random matrix theory and differential geometry. We generalize our hypercontractive inequalities from $SU(n)$ to an arbitrary $D$-quasirandom compact connected Lie group for $D$ at least an absolute constant, thereby extending our results on product-free sets to such groups.
We also demonstrate various other applications of our inequalities to geometry (viz., non-Abelian Brunn-Minkowski type inequalities), mixing times, and the theory of growth in compact Lie groups. A subsequent work due to Arunachalam, Girish and Lifshitz uses our inequalities to establish new separation results between classical and quantum communication complexity.Wed, 13 Sep 2023 16:13:12 +0300https://eccc.weizmann.ac.il/report/2023/133