A NEW PROOF OF SZEMER ´ EDI'S THEOREM

By William T. Gowers. 2001

428 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/3eb1c044a5fbf5d03a016478aae6b24df98d917a

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

Citations in the corpus

Pointwise convergence of ergodic averages along cubes (2010, background): “These expressions can be viewed as discrete versions of continuous averages introduced by T. Gowers [ 10 ].”
Higher moments of convolutions (2011, methodology): “lso given in the section 9). The results are particularly powerful in the case of multiplicative subgroups of the ﬁeld Fq. In section 9 we prove two versions of the well–known Balog–Szemer´edi–Gowers [2, 4, 11]. Assuming E2(A) = |A|3/K and E3(A) 6 M|A|4/K2 we obtain an improvement of Balog– Szemer´e”
Combinatorial theorems in sparse random sets (2010, background): “on p= p(n) tending to zero with nsuch that lim n→∞ P([n]p is (δ,k)-Szemer´edi) = 1. The approach of Green and Tao depends heavily on the use of a set of norms known as uniformity norms, introduced in [19]. In order to deal with k-term arithmetic progressions, one must use a uniformity norm that is b”
Growth in groups: ideas and perspectives (2013, unknown): “etic progressions by Schur, van der Waerden, Roth [Rot53], Szemer´edi [Sze69], Furstenberg [Fur77] (leading to the ergodic work of Host-Kra [HK05] and Ziegler [Zie07]; see also Szegedy [Sze]), Gowers [Gow01], and Green-Tao [GT08], among many others, and another one based on the study of growth in ab”
COMPRESSIONS, CONVEX GEOMETRY AND THE FREIMAN–BILU THEOREM (2005, methodology): “is necessary to have the Freiman-Bilu theorem for arbitrarily small ε. For many applications in additive combinatorics (for example to the problem of obtaining inverse theorems for the Gowers U3-norm [9, 13] or to the sum-product problem [4]) we would be just as happy with ε = 1, or even to permit P”
Graph norms and Sidorenko’s conjecture (2008, background): “e primes contain arbitrarily long arithmetic progressions and the current best bounds for the quantiﬁed version of the Szemer´edi’s theorem is through the so called “inverse theorems” for these norms [15, 16, 17, 14]. With all the known applications for the Gowers norms, it seems natural to believe ”
An improved point‐line incidence bound over arbitrary fields (2016, methodology): “heorems 3 and 4 to be optimal. Previous approaches to incidence problems over ﬁnite ﬁelds relied on techniques from additive combinatorics, and in particular the Balog-Szemer´edi-Gowers (BSG) theorem [10]. A number of recent papers, including [26, 1, 28], have successfully replaced the traditional a”
Linear Forms and Higher-Degree Uniformity for Functions On $${\mathbb{F}^{n}_{p}}$$ (2010, background): “ǫ.h) ≤ Y ǫ∈{0,1}k+1 kfǫkUk+1. Here C|ǫ| is the operation of taking the complex conjugate |ǫ| times. A proof of this inequality (for ZN rather than Fn p, but the argument is identical) can be found in [G01]. We apply this result with f0 = f and fǫ = g= ωπ for every other ǫ∈ {0,1}k+1. The identity P ǫ”
A new proof of the density Hales-Jewett theorem (2009, background): “let us sketch the argument that gives us corners. The basic idea is a density increment strategy, which has been used to prove many density theorems. (A few examples can be found in [Rot53], [Sze75], [Gow01], [Shk06b], [Shk06a], and [LM08], but this is by no means an exhaustive list.) We shall show ”
A transference approach to a Roth-type theorem in the squares (2015, background): “,a) q ∈ K. A ROTH-TYPE THEOREM IN THE SQUARES 5 Thus A cannot contain an arithmetic progression of length k, so that |A| ≪ X(loglogX)−ck,for some c k >0, by Gowers’s bounds in Szemer´edi’s theorem [Gow01]. This argument covers those equations of the form (1.4) tackled by Smith, Keil and Henriot, ”