50 research outputs found
Low-degree tests at large distances
We define tests of boolean functions which distinguish between linear (or
quadratic) polynomials, and functions which are very far, in an appropriate
sense, from these polynomials. The tests have optimal or nearly optimal
trade-offs between soundness and the number of queries.
In particular, we show that functions with small Gowers uniformity norms
behave ``randomly'' with respect to hypergraph linearity tests.
A central step in our analysis of quadraticity tests is the proof of an
inverse theorem for the third Gowers uniformity norm of boolean functions.
The last result has also a coding theory application. It is possible to
estimate efficiently the distance from the second-order Reed-Muller code on
inputs lying far beyond its list-decoding radius
An inverse theorem for the Gowers U^3 norm
The Gowers U^3 norm is one of a sequence of norms used in the study of
arithmetic progressions. If G is an abelian group and A is a subset of G then
the U^3(G) of the characteristic function 1_A is useful in the study of
progressions of length 4 in A. We give a comprehensive study of the U^3(G)
norm, obtaining a reasonably complete description of functions f : G -> C for
which ||f||_{U^3} is large and providing links to recent results of Host, Kra
and Ziegler in ergodic theory.
As an application we generalise a result of Gowers on Szemeredi's theorem.
Writing r_4(G) for the size of the largest set A not containing four distinct
elements in arithmetic progression, we show that r_4(G) << |G|(loglog|G|)^{-c}
for some absolute constant c.
In future papers we will develop these ideas further, obtaining an asymptotic
for the number of 4-term progressions p_1 < p_2 < p_3 < p_4 < N of primes as
well as superior bounds for r_4(G).Comment: 72 pages, some spelling corrections and updated references. To appear
in Proc. Edinburgh Math. So
Quantitative bounds for Gowers uniformity of the M\"obius and von Mangoldt functions
We establish quantitative bounds on the Gowers norms of the M\"obius
function and the von Mangoldt function for all , with error
terms of shape . As a consequence, we obtain quantitative
bounds for the number of solutions to any linear system of equations of finite
complexity in the primes, with the same shape of error terms. We also obtain
the first quantitative bounds on the size of sets containing no -term
arithmetic progressions with shifted prime difference.Comment: 56 pages; added Corollary 1.5 and Theorem 1.
An equivalence between inverse sumset theorems and inverse conjectures for the U^3 norm
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
polynomials). In particular, we show that the inverse sumset theorems of
Freiman type are equivalent to the known inverse results for the Gowers U^3
norms, and moreover that the conjectured polynomial strengthening of the former
is also equivalent to the polynomial strengthening of the latter. We establish
this equivalence in two model settings, namely that of the finite field vector
spaces F_2^n, and of the cyclic groups Z/NZ.
In both cases the argument involves clarifying the structure of certain types
of approximate homomorphism.Comment: 23 page
Long arithmetic progressions of primes
This is an article for a general mathematical audience on the author's work,
joint with Terence Tao, establishing that there are arbitrarily long arithmetic
progressions of primes.
It is based on several one hour lectures, chiefly given at British
universities.Comment: 19 pages, submitted to Proceedings of the Gauss-Dirichlet Conference,
Gottingen, June 20-24 200