Guy Kindler
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View article: Product Mixing in Compact Lie Groups
Product Mixing in Compact Lie Groups Open
If 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 ∈ S.In 1985, Babai and Sós [] asked, for a finite group G, how large a subset S⊆ G can be if it is product-free. The main tool (hither…
View article: Limits of Preprocessing
Limits of Preprocessing Open
It is a classical result that the inner product function cannot be computed by an $${\rm AC}^0$$ circuit. It is conjectured that this holds even if we allow arbitrary preprocessing of each of the two inputs separately. We prove this c…
View article: Polynomial Bogolyubov for special linear groups via tensor rank
Polynomial Bogolyubov for special linear groups via tensor rank Open
We prove a polynomial Bogolyubov type lemma for the special linear group over finite fields. Specifically, we show that there exists an absolute constant $C>0,$ such that if $A$ is a density $α$ subset of the special linear group, then the…
View article: Product Mixing in Compact Lie Groups
Product Mixing in Compact Lie Groups Open
If $G$ 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 ir…
View article: Hypercontractivity on the symmetric group
Hypercontractivity on the symmetric group Open
The hypercontractive inequality is a fundamental result in analysis, with many applications throughout discrete mathematics, theoretical computer science, combinatorics and more. So far, variants of this inequality have been proved mainly …
View article: An Analogue of Bonami’s Lemma for Functions on Spaces of Linear Maps, and 2-2 Games
An Analogue of Bonami’s Lemma for Functions on Spaces of Linear Maps, and 2-2 Games Open
We prove an analogue of Bonami's (hypercontractive) lemma for complex-valued functions on L (𝑉 ,𝑊 ), where 𝑉 and 𝑊 are vector spaces over a finite field. This inequality is useful for functions on L (𝑉 ,𝑊 ) whose 'generalised influences' a…
View article: Improved Monotonicity Testers via Hypercube Embeddings
Improved Monotonicity Testers via Hypercube Embeddings Open
We show improved monotonicity testers for the Boolean hypercube under the $p$-biased measure, as well as over the hypergrid $[m]^n$. Our results are: 1. For any $p\in (0,1)$, for the $p$-biased hypercube we show a non-adaptive tester that …
View article: An analogue of Bonami's Lemma for functions on spaces of linear maps, and 2-2 Games
An analogue of Bonami's Lemma for functions on spaces of linear maps, and 2-2 Games Open
We prove an analogue of Bonami's (hypercontractive) lemma for complex-valued functions on $\mathcal{L}(V,W)$, where $V$ and $W$ are vector spaces over a finite field. This inequality is useful for functions on $\mathcal{L}(V,W)$ whose `gen…
View article: The success probability in Levine's hat problem, and independent sets in graphs
The success probability in Levine's hat problem, and independent sets in graphs Open
Lionel Levine's hat challenge has $t$ players, each with a (very large, or infinite) stack of hats on their head, each hat independently colored at random black or white. The players are allowed to coordinate before the random colors are c…
View article: Forbidden intersection problems for families of linear maps
Forbidden intersection problems for families of linear maps Open
We study an analogue of the Erdős-Sós forbidden intersection problem, for families of linear maps. If $V$ and $W$ are vector spaces over the same field, we say a family $\mathcal{F}$ of linear maps from $V$ to $W$ is \emph{$(t-1)$-intersec…
View article: Isoperimetric Inequalities Made Simpler
Isoperimetric Inequalities Made Simpler Open
We give an alternative, simple method to prove isoperimetric inequalities over the hypercube. In particular, we show: 1. An elementary proof of classical isoperimetric inequalities of Talagrand, as well as a stronger isoperimetric result c…
View article: owards a quantum-inspired proof for IP = PSPACE
owards a quantum-inspired proof for IP = PSPACE Open
We explore quantum-inspired interactive proof systems where the prover is limited. Namely, we improve on a result by \cite{AG17} showing a quantum-inspired interactive protocol ($\IP$) for $PreciseBQP$ where the prover is only assumed to b…
View article: The success probability in Lionel Levine's hat problem is strictly decreasing with the number of players, and this is related to interesting questions regarding Hamming powers of Kneser graphs and independent sets in random subgraphs
The success probability in Lionel Levine's hat problem is strictly decreasing with the number of players, and this is related to interesting questions regarding Hamming powers of Kneser graphs and independent sets in random subgraphs Open
Lionel Levine's hat challenge has $t$ players, each with a (very large, or infinite) stack of hats on their head, each hat independently colored at random black or white. The players are allowed to coordinate before the random colors are c…
View article: Theorems of KKL, Friedgut, and Talagrand via Random Restrictions and Log-Sobolev Inequality
Theorems of KKL, Friedgut, and Talagrand via Random Restrictions and Log-Sobolev Inequality Open
We give alternate proofs for three related results in analysis of Boolean functions, namely the KKL Theorem, Friedgut’s Junta Theorem, and Talagrand’s strengthening of the KKL Theorem. We follow a new approach: looking at the first Fourier…
View article: Limits of Preprocessing
Limits of Preprocessing Open
It is a classical result that the inner product function cannot be computed by an AC⁰ circuit [Merrick L. Furst et al., 1981; Miklós Ajtai, 1983; Johan Håstad, 1986]. It is conjectured that this holds even if we allow arbitrary preprocessi…
View article: Concentration on the Boolean hypercube via pathwise stochastic analysis
Concentration on the Boolean hypercube via pathwise stochastic analysis Open
We develop a new technique for proving concentration inequalities which\nrelate between the variance and influences of Boolean functions. Using this\ntechnique, we\n 1. Settle a conjecture of Talagrand [Tal97] proving that $$\\int_{\\left\…
View article: Invariance Principle on the Slice
Invariance Principle on the Slice Open
The non-linear invariance principle of Mossel, O’Donnell, and Oleszkiewicz establishes that if f ( x 1 ,… , x n ) is a multilinear low-degree polynomial with low influences, then the distribution of if f ( b 1 ,…, b n ) is close (in variou…
View article: Quantum automata cannot detect biased coins, even in the limit
Quantum automata cannot detect biased coins, even in the limit Open
Aaronson and Drucker (2011) asked whether there exists a quantum finite automaton that can distinguish fair coin tosses from biased ones by spending significantly more time in accepting states, on average, given an infinite sequence of tos…
View article: Geometric stability via information theory
Geometric stability via information theory Open
The Loomis-Whitney inequality, and the more general Uniform Cover inequality, bound the volume of a body in terms of a product of the volumes of lower-dimensional projections of the body. In this paper, we prove stability versions of these…