A BILINEAR VERSION OF BOGOLYUBOV'S THEOREM

A theorem of Bogolyubov states that for every dense set A A in Z N \mathbb {Z}_N we may find a large Bohr set inside A + A − A − A A+A-A-A . In this note, motivated by work on a quantitative inverse theorem for the Gowers U 4 U^4 norm, we prove a bilinear variant of this result for vector spaces over finite fields. Given a subset A ⊂ F p n × F p n A \subset \mathbb {F}^n_p \times \mathbb {F}^n_p , we consider two operations: one of them replaces each row of A A by the set difference of it with itself, and the other does the same for columns. We prove that if A A has positive density and these operations are repeated several times, then the resulting set contains a bilinear analogue of a Bohr set, namely the zero set of a biaffine map from F p n × F p n \mathbb {F}^n_p \times \mathbb {F}^n_p to an F p \mathbb {F}_p -vector space of bounded dimension. An almost identical result was proved independently by Bienvenu and Lê.This is funded by my Royal Society Research Professorshi

INTERLEAVED GROUP PRODUCTS

Let $G$ be the special linear group $\mathrm{SL}(2,q)$. We show that if $(a_1,\ldots,a_t)$ and $(b_1,\ldots,b_t)$ are sampled uniformly from large subsets $A$ and $B$ of $G^t$ then their interleaved product $a_1 b_1 a_2 b_2 \cdots a_t b_t$ is nearly uniform over $G$. This extends a result of the first author, which corresponds to the independent case where $A$ and $B$ are product sets. We obtain a number of other results. For example, we show that if $X$ is a probability distribution on $G^m$ such that any two coordinates are uniform in $G^2$, then a pointwise product of $s$ independent copies of $X$ is nearly uniform in $G^m$, where $s$ depends on $m$ only. Extensions to other groups are also discussed. We obtain closely related results in communication complexity, which is the setting where some of these questions were first asked by Miles and Viola. For example, suppose party $A_i$ of $k$ parties $A_1,\dots,A_k$ receives on its forehead a $t$-tuple $(a_{i1},\dots,a_{it})$ of elements from $G$. The parties are promised that the interleaved product $a_{11}\dots a_{k1}a_{12}\dots a_{k2}\dots a_{1t}\dots a_{kt}$ is equal either to the identity $e$ or to some other fixed element $g\in G$, and their goal is to determine which of the two the product is equal to. We show that for all fixed $k$ and all sufficiently large $t$ the communication is $\Omega(t \log |G|)$, which is tight. Even for $k=2$ the previous best lower bound was $\Omega(t)$. As an application, we establish the security of the leakage-resilient circuits studied by Miles and Viola in the "only computation leaks" model

Freiman homomorphisms on sparse random sets

A result of Fiz Pontiveros shows that if $A$ is a random subset of $\mathbb{Z}_N$ where each element is chosen independently with probability $N^{-1/2+o(1)}$, then with high probability every Freiman homomorphism defined on $A$ can be extended to a Freiman homomorphism on the whole of $\mathbb{Z}_N$. In this paper we improve the bound to $CN^{-2/3}(\log N)^{1/3}$, which is best possible up to the constant factor.Research supported by a Royal Society University Research Fellowship and by ERC Starting Grant 676632. Research supported by a Royal Society 2010 Anniversary Research Professorship

Inverse and stability theorems for approximate representations of finite groups

The U 2 norm gives a useful measure of quasirandomness for realor complex-valued functions defined on finite (or, more generally, locally compact) groups. A simple Fourier-analytic argument yields an inverse theorem, which shows that a bounded function with a large U 2 norm defined on a finite Abelian group must correlate significantly with a character. In this paper we generalize this statement to functions that are defined on arbitrary finite groups and that take values in Mn(C). The conclusion now is that the function correlates with a representation – though with the twist that the dimension of the representation is shown to be within a constant of n rather than being exactly equal to n. There are easy examples that show that this weakening of the obvious conclusion is necessary. The proof is much less straightforward than it is in the case of scalar functions on Abelian groups. As an easy corollary, we prove a stability theorem for near representations. It states that if G is a finite group and f : G →Mn(C) is a function that is close to a representation in the sense that f(xy) − f(x)f(y) has a small Hilbert-Schmidt norm (also known as the Frobenius norm) for every x, y ∈ G, then there must be a representation ρ such that f(x) − ρ(x) has small Hilbert-Schmidt norm for every x. Again, the dimension of ρ need not be exactly n, but it must be close to n. We also obtain stability theorems for other Schatten p-norms. A stability theorem of this kind was obtained for the operator norm by Grove, Karcher and Ruh in 1974 [6] and in a more general form by Kazhdan in 1982 [8]. (For the operator norm, the dimension of the approximating representation is exactly n.

Knowledge management in future organizations

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Diversity in proof appraisal

We investigated whether mathematicians typically agree about the qualities of mathematical proofs. Between-mathematician consensus in proof appraisals is an implicit assumption of many arguments made by philosophers of mathematics, but to our knowledge the issue has not previously been empirically investigated. We asked a group of mathematicians to assess a specific proof on four dimensions, using the framework identified by Inglis and Aberdein (2015). We found widespread disagreement between our participants about the aesthetics, intricacy, precision and utility of the proof, suggesting that a priori assumptions about the consistency of mathematical proof appraisals are unreasonable

Combinatorial theorems in sparse random sets

We develop a new technique that allows us to show in a unified way that many well-known combinatorial theorems, including Turán’s theorem, Szemerédi’s theorem and Ramsey’s theorem, hold almost surely inside sparse random sets. For instance, we extend Turán’s theorem to the random setting by showing that for every ϵ>0 and every positive integer t≥3 there exists a constant C such that, if G is a random graph on n vertices where each edge is chosen independently with probability at least Cn−2/(t+1), then, with probability tending to 1 as n tends to infinity, every subgraph of G with at least (1–1t−1+ϵ)e(G) edges contains a copy of Kt. This is sharp up to the constant C. We also show how to prove sparse analogues of structural results, giving two main applications, a stability version of the random Turán theorem stated above and a sparse hypergraph removal lemma. Many similar results have recently been obtained independently in a different way by Schacht and by Friedgut, Rödl and Schacht

Improved bounds for the Erdős-Rogers function

The Erd\H{o}s-Rogers function $f_{s,t}$ measures how large a $K_s$-free induced subgraph there must be in a $K_t$-free graph on $n$ vertices. While good estimates for $f_{s,t}$ are known for some pairs $(s,t)$, notably when $t=s+1$, in general there are significant gaps between the best known upper and lower bounds. We improve the upper bounds when $s+2\leq t\leq 2s-1$. For each such pair we obtain for the first time a proof that $f_{s,t}\leq n^{\alpha_{s,t}+o(1)}$ with an exponent $\alpha_{s,t}<1/2$, answering a question of Dudek, Retter and R\"{o}dl