Related papers: Inverse and stability theorems for approximate rep…

We prove an inverse theorem for the Gowers $U^2$-norm for maps $G\to\mathcal M$ from an countable, discrete, amenable group $G$ into a von Neumann algebra $\mathcal M$ equipped with an ultraweakly lower semi-continuous, unitarily invariant…

Classical and recent results on uncertainty principles for functions on finite Abelian groups relate the cardinality of the support of a function to the cardinality of the support of its Fourier transforms. We use these results and their…

In this paper, we develop a quantitative inverse theory for the Gowers uniformity norm $\|\cdot\|_{\mathsf{U}^4}$ in general finite abelian groups. We identify a new type of obstructions to uniformity, which we call almost-cubic…

For every natural number k we prove a decomposition theorem for bounded measurable functions on compact abelian groups into a structured part, a quasi random part and a small error term. In this theorem quasi randomness is measured with the…

A very useful fact in additive combinatorics is that analytic expressions that can be used to count the number of structures of various kinds in subsets of Abelian groups are robust under quasirandom perturbations, and moreover that…

Gowers and Hatami initiated the inverse theory for the uniformity norms $U^k$ of matrix-valued functions on non-abelian groups by proving a $1\%$-inverse theorem for the $U^2$-norm and relating it to stability questions for almost…

If matrices almost satisfying a group relation are close to matrices exactly satisfying the relation, then we say that a group is matricially stable. Here "almost" and "close" are in terms of the Hilbert-Schmidt norm. Using tracial 2-norm…

This paper discusses a general and useful stability principle which, roughly speaking, says that given a uniformly continuous function defined on an arbitrary metric space, if the function is bounded on the constraint set and we slightly…

A result of the author shows that the behavior of Gowers norms on bounded exponent abelian groups is connected to finite nilspaces. Motivated by this, we investigate the structure of finite nilspaces. As an application we prove inverse…

We establish inverse and direct theorems on best approximations in quasi-normed Abelian groups through bilateral Bernstein-Jackson inequalities with exact constants. Using integral representations for quasi-norms of functions $f$ in…

We state and prove a quantitative inverse theorem for the Gowers uniformity norm $U^3(G)$ on an arbitrary finite abelian group $G$; the cases when $G$ was of odd order or a vector space over ${\mathbf F}_2$ had previously been established…

Approximate algebraic structures play a defining role in arithmetic combinatorics and have found remarkable applications to basic questions in number theory and pseudorandomness. Here we study approximate representations of finite groups:…

We establish a correspondence between inverse sumset theorems (which can be viewed as classifications of approximate (abelian) groups) and inverse theorems for the Gowers norms (which can be viewed as classifications of approximate…

We prove a structure theorem for stable functions on amenable groups, which extends the arithmetic regularity lemma for stable subsets of finite groups. Given a group $G$, a function $f\colon G\to [-1,1]$ is called stable if the binary…

We introduce a relaxation of stability, called almost sure stability, which is insensitive to perturbations by subsets of Loeb measure $0$ in a non-standard finite group. We show that almost sure stability satisfies a stationarity principle…

We prove an inversion theorem for the Fourier transform defined for normal functions, in the case when such functions are of moderate decrease, and in dimensions 2 and 3. This improves on Carleson's general almost everywhere convergence…

We give a structural description of the finite subsets $A$ of an arbitrary group $G$ which obey the polynomial growth condition $|A^n| \leq n^d |A|$ for some bounded $d$ and sufficiently large $n$, showing that such sets are controlled by…

Motivated by expansion in Cayley graphs, we show that there exist infinitely many groups $G$ with a nontrivial irreducible unitary representation whose average over every set of $o(\log\log|G|)$ elements of $G$ has operator norm $1 - o(1)$.…

A combination of direct and inverse Fourier transforms on the unitary group $U(N)$ identifies normalized characters with probability measures on $N$-tuples of integers. We develop the $N\to\infty$ version of this correspondence by matching…

We establish the \emph{inverse conjecture for the Gowers norm over finite fields}, which asserts (roughly speaking) that if a bounded function $f: V \to \C$ on a finite-dimensional vector space $V$ over a finite field $\F$ has large Gowers…