# Quantitative bounds in the inverse theorem for the Gowers $U^{s+1}$-norms over cyclic groups

@article{Manners2018QuantitativeBI, title={Quantitative bounds in the inverse theorem for the Gowers \$U^\{s+1\}\$-norms over cyclic groups}, author={Frederick Manners}, journal={arXiv: Combinatorics}, year={2018} }

We provide a new proof of the inverse theorem for the Gowers $U^{s+1}$-norm over groups $H=\mathbb Z/N\mathbb Z$ for $N$ prime. This proof gives reasonable quantitative bounds (the worst parameters are double-exponential), and in particular does not make use of regularity or non-standard analysis, both of which are new for $s \ge 3$ in this setting.

## 9 Citations

The inverse theorem for the $U^3$ Gowers uniformity norm on arbitrary finite abelian groups: Fourier-analytic and ergodic approaches

- Mathematics
- 2021

We state and prove a quantitative inverse theorem for the Gowers uniformity norm U(G) on an arbitrary finite group G; the cases when G was of odd order or a vector space over F2 had previously been…

Quantitative bounds for Gowers uniformity of the M\"obius and von Mangoldt functions

- Mathematics
- 2021

We establish quantitative bounds on the U[N ] Gowers norms of the Möbius function μ and the von Mangoldt function Λ for all k, with error terms of shapeO((log logN)−c). As a consequence, we obtain…

Quantitative inverse theory of Gowers uniformity norms

- Mathematics
- 2020

(This text is a survey written for the Bourbaki seminar on the work of F. Manners.)
Gowers uniformity norms are the central objects of higher order Fourier analysis, one of the cornerstones of…

Partition Rank and Analytic Rank are Uniformly Equivalent

- Mathematics
- 2021

We prove that the partition rank and the analytic rank of tensors are equal up to a constant, over finite fields of any characteristic and any large enough cardinality, independently of the number of…

An inverse theorem for Freiman multi-homomorphisms

- Mathematics
- 2020

Let $G_1, \dots, G_k$ and $H$ be vector spaces over a finite field $\mathbb{F}_p$ of prime order. Let $A \subset G_1 \times\dots\times G_k$ be a set of size $\delta |G_1| \cdots |G_k|$. Let a map…

Certain Directional Gowers Uniformity Norms achieved by Manners in [ 26 ]

- Mathematics
- 2021

which is the simplest interesting unknown case of the inverse problem for the directional Gowers uniformity norms. Namely, writing ‖·‖U for the norm above, we show that if f :G → C is a function…

Regularity and inverse theorems for uniformity norms on compact abelian groups and nilmanifolds

- Mathematics
- 2019

We prove a general form of the regularity theorem for uniformity norms, and deduce a generalization of the Green-Tao-Ziegler inverse theorem, extending it to a class of compact nilspaces including…

Approximately Symmetric Forms Far From Being Exactly Symmetric

- Mathematics
- 2021

Let V be a finite-dimensional vector space over Fp. We say that a multilinear form α:V k → Fp in k variables is d-approximately symmetric if the partition rank of difference α(x1, . . . , xk) −…

## References

SHOWING 1-10 OF 13 REFERENCES

Periodic nilsequences and inverse theorems on cyclic groups

- Mathematics
- 2014

The inverse theorem for the Gowers norms, in the form proved by Green, Tao and Ziegler, applies to functions on an interval $[M]$. A recent paper of Candela and Sisask requires a stronger conclusion…

The quantitative behaviour of polynomial orbits on nilmanifolds

- Mathematics
- 2007

A theorem of Leibman asserts that a polynomial orbit $(g(1),g(2),g(3),\ldots)$ on a nilmanifold $G/\Gamma$ is always equidistributed in a union of closed sub-nilmanifolds of $G/\Gamma$. In this paper…

On the Bogolyubov–Ruzsa lemma

- Mathematics
- 2012

Our main result is that if A is a finite subset of an abelian group with |A+A| < K|A|, then 2A-2A contains an O(log^{O(1)} K)-dimensional coset progression M of size at least exp(-O(log^{O(1)} K))|A|.

A new proof of Szemerédi's theorem

- Mathematics
- 2001

In 1927 van der Waerden published his celebrated theorem on arithmetic progressions, which states that if the positive integers are partitioned into finitely many classes, then at least one of these…

Extending linear and quadratic functions from high rank varieties

- Mathematics
- 2017

Let $k$ be a field, $V$ be a $k$-vector space and $X\subset V$ an algebraic irreducible subvariety.
We say that a function $f:X(k) \to k$ is weakly linear if its restriction to any two-dimensional…

Linear equations in primes

- Mathematics
- 2006

Consider a system ψ of nonconstant affine-linear forms ψ 1 , ... , ψ t : ℤ d → ℤ, no two of which are linearly dependent. Let N be a large integer, and let K ⊆ [-N, N] d be convex. A generalisation…

Distribution of values of bounded generalized polynomials

- Mathematics
- 2007

A generalized polynomial is a real-valued function which is obtained from conventional polynomials by the use of the operations of addition, multiplication, and taking the integer part; a generalized…

The true complexity of a system of linear equations

- Mathematics
- 2010

In this paper we look for conditions that are sufficient to guarantee that a subset A of a finite Abelian group G contains the ‘expected’ number of linear configurations of a given type. The simplest…

Testing Low-Degree Polynomials over GF(2(

- Mathematics, Computer ScienceRANDOM-APPROX
- 2003

Any algorithm for testing degree-k polynomials over GF(2) must perform \(\Omega(\frac{1}{\epsilon}+2^k)\) queries, and the result is essentially tight.

Notes on compact nilspaces

- Mathematics
- 2016

These notes form the second part of a detailed account of the theory of nilspaces developed by Camarena and Szegedy. Here we focus on nilspaces equipped with a compact topology that is compatible…