RESEARCH STATEMENT
TATCHAI TITICHETRAKUN
1. I NTRODUCTION
My primary research interest is in the area of additive combinatorics. My work was motivated
by the study of additive structure in dense subsets of integers, in particular, affine copies of a
finite configuration. Many classical results on patterns in dense sets can be studied via classical
Fourier analysis, but for some more complicated patterns, we will need more advanced tools. In
Sections 3.1, 3.2, 3.3, I note some of my results in this direction. My main motivation comes from
trying to understand the structural objects (could be described in terms of Gowers uniformity norm,
nilsequences, see Section 2) that play a role in revealing these complicated patterns in dense sets.
Their properties and roles in additive combinatorics are still not perfectly understood, even in the
proof of Inverse Gowers Norm conjecture [28]. Also, as a relatively new concept, it is interesting to
study their relationships and new applications both directly in the fields of additive combinatorics,
ergodic theory, harmonic analysis or number theory to other fields like computer science ( see e.g.
the first section of the survey [52] for some examples).
2. BACKGROUND
Recent developments in additive combinatorics come from the following Ramsey type theorems
where not only the statements that are interesting but also many of their proof methods. Now, there
are many approaches to attack these problems on more complicated patterns including multiple recurrence in ergodic theory, regularity of hypergraph and additive number theory. These approaches
could be considered as a part of higher order Fourier analysis.
Proposition 1 (Szemerédi (d=1 [45]),
P Furstenberg-Katznelson (d ≥ 1 [11])). Let f be a function
supported on [1, N ]d and (1/N d ) x∈[1,N ]d f (x) ≥ α for some 0 < α < 1. Then for any given
finite set F = {v1 , . . . , vk } ⊆ Zd , we have
X
1
f (x)f (x + tv1 )f (x + tv2 ) . . . f (x + tvk ) ≥ c(α, F ) > 0
(1)
N d+1
d
x∈Z ,t∈Z
Note that c(α, F ) is independent of f .
As a corollary, taking f to be the characteristic function of A, it is not hard to find a nontrivial
affine copy of F i.e. we can find x ∈ Zd , t ∈ Z such that x + tF ∈ A. A general strategy of
proving theorems like Proposition 1 is to decompose any bounded function f as
f = f1 + f2 + f3
(2)
where f1 has some algebraic structure which is easy to study or use to prove an estimate like (1).
f2 is a part that does not have such structure and is expected to behave randomly that causes lots of
cancellations and contributes only a small error term in (1). f3 is an error term that is, for example,
Date: December 10 2015.
1
small in L2 norm. An example of this strategy is regularity lemma (see e.g. Proposition 2) first
appeared in the work of Szemerédi [45] in the context of graph theory.
The case d = 1, k = 3 of Proposition 1 is first proved by Roth [41]. He described f2 as a
function with small Fourier transform i.e. functions that do not correlate with any exponentials
of linear phase functions f (x) = eirx . Hence a structure theorem which gives a decomposition
like (2) is quite easy to describe in this case. However for k = 4, it turns out that we need more
objects other than the linear exponentials above to give a good description of f2 ; this is essentially
because of the quadratic correlation x2 − 3(x + d)2 + 3(x + 2d)2 − (x + 3d)2 = 0 arisen in 4-term
arithmetic progressions which could not be captured by the linear phases. In Gowers’s proof [14],
[15] of the case k ≥ 4, he introduced the Gowers uniformity norm to measure uniformity. Gowers
uniformity norm of degree 2 of f : ZN → C is defined as
X
1
kf k4U 2 := 3
f (x)f (x + h1 )f (x + h2 )f (x + h1 + h2 )
N x,h ,h ∈Z
1
2
N
Equivalently, this is the average over quadruples (u, v, z, w) such that u + w = z + v which forms
a parallelogram in dimension 2 (this relates the cubic structures in dynamics). This definition of
U 2 norm is easy to generalize to higher U k . It is easy to show
kf kU 2 = kfˆk4
(3)
This gives the term higher order Fourier analysis, however it is still unknown whether there is a
formula analogue to the identity (3) in the case k > 2 where nilsequences play a role of the linear
characters. The key point is that functions with large Gowers norm of some degree will correlate
with some set of structural objects of that degree. A precise description of these objects on ZN
are described by the Inverse Theorem of Gowers Norm of Green-Tao-Ziegler [28]. Higher order
Gowers norm could capture some more complicated structure in a set or function that may not be
caught by lower Gowers norms.
In [12] and [38] Gowers and independently, Nagel-Rödl-Schacht give a new proof of Proposition
1 including the higher dimension case using regularity on (k + 1)−partite k−regular hypergraphs
initiated by a simple idea of Solymosi [44]. Here we state an example of hypergraph regularity
for 3-regular 3-partite hypergraphs with finite non-empty vertex sets X, Y, Z. Recall that a sigmaalgebra B on a finite set is given by a unique partition of that finite set . The conditional expectation
E(f |B) can be considered as information of f that is captured by the sigma-algebra (partition) B.
Box norm kf k3 is the analogue of Gowers uniformity norm kf kU 3 on 3-regular hypergraph. If a
function f has kf − E(f |B)k3 small then we can say that all structures of f relating to 3 −norm
(in the sense of the inverse theorem) could be captured by B.
Proposition 2 (Hypergraph Regularity Lemma). Let f : X × Y × Z → [0, 1]. Let > 0 and
F : R → R be arbitrary quickly increasing function possibly depends on , then there are sigmaalgebras BX,Y , BX,Z , BY,Z on X × Y, X × Z, Y × Z respectively of complexity at most M such
that f = f1 + f2 + f3 where f1 = E(f |BX,Y ∨ BY,Z ∨ BX,Z ), kf2 k3 ≤ F (M )−1 , kf3 k2 ≤ .
Hence this theorem roughly says that the structure 3- regular hypergraph on vertices set X ×
Y × Z can be approximately described in terms of lower order edges between (X, Y ) and (Y, Z)
and (Z, X). A feature of regularity lemma is that we have arbitrary accurate control on uniformity
norm kf2 k in terms of the complexity M . However, it is shown by Gowers [16] that we can
only obtain a iterated power type bound in (1) from regularity lemma (the allowance of arbitrary
2
increasing functions put the discrete problem to a linkage with related infinitary regime like ergodic
theory) . Another decomposition strategy may be needed for a better quantitative bound.
In the context of ergodic theory, Furstenberg [11] introduced Furstenberg’s Corresponding Principle that to study Proposition 1 (in case d = 1, for example), it is enough to study the following
multiple recurrence in a probability measure preserving system X.
N
1 X
f1 (T n x)f2 (T 2n x) . . . fk (T kn x)
N n=1
(4)
Here fi ∈ L∞ (X) and T : X → X is measure preserving. In proving limiting behavior of (4),
there is an idea of using the characteristic factor Z which we may think of as invariant subsigmaalgebra Z such that if E(fi |Z) = 0 for some i then the limit of (4) would be 0 in L2 norm. This is in
a similar spirit of decomposing functions as in (2). The question of finding the characteristic factor
for a given multiple recurrence is a delicate one. Host and Kra [30] and Ziegler [54] are able to give
a nice description of the characteristic factor of (4) in terms of nilrotation on nilmanifolds (could
be considered as a generalization of abelian rotation on S 1 ) and Host-Kra seminorm (an analogue
of Gowers norm). Nilsequences [2] play a role as the obstruction to uniformity similar to linear
exponentials in Roth’s theorem case. This motivated parallel work in additive combinatorics.
2.1. Quantitative Questions. Most results in density Ramsey theory (especially in higher dimensions) could only be attacked by ergodic theory which does not give a good quantitative bound on
the number of configurations. More understanding in higher order Fourier analysis is still needed
for other approaches, for example, we don’t have a good systematic treatment to treat the theory of
arithmetic progression of length ≥ 5. Getting good bounds on theorems like Gowers Inverse Norm
Theorem [28] (and its generalizations to higher dimensions or more general (high rank) abelian
groups) and its relations to additive theory of sets like , Frieman’s Theorem (structure of finite sets
with small doubling, see e.g. [40]) or approximate group ( see e.g. [20], which may give a new
approach to inverse U 3 -theorem), is a major open problem in this area. Solutions to such problems
would come with better understanding of the subject.
In general, it is convenient to work on a finite group like Z/pZ to get results on Z. It is also
more convenient to work on Fnp for large n (known as finite field model) where some vector space
substructures are available. These substructures are not available in Z/pZ. For example, Green
and Tao [26] count (with improved bound) arithmetic progression of length 4 in finite field model
before applying the idea obtained from finite field model modified with some technical tools to
obtain results in Z [27]. The idea in [26] is to decompose function locally on a subspace (with some
rank conditions) rather than do it globally. Then we can obtain a better bound in the decomposition
(2) locally. It would be interesting to find applications of their local k · kU 3 theory in other settings,
for example, corners in dimension 3 in the finite field setting (exponential bound only known in
the case of dimension 2 due to Shkredov [43]). Apart from the finite field setting, there is function
field setting Fq (t)/hpi where p is an irreducible polynomial of degree n and q = pn which can
provides finite character analogue of {1, . . . , N = q n }. This setting works well with both addition
and multiplication. There are results on Green-Tao’s Theorem [32], Roth’s theorem [3] and sumproduct problem [4] where the bound could be better in function field setting. Since we also have
Bohr sets (analogue of subspaces) in this setting, it would be interesting to develop k · kU 3 theory
in this setting which are related to, say, arithmetic progressions of length 4.
3
3. PATTERNS IN THE P RIMES
Denote by P, PN the set of primes and the set of primes ≤ N respectively. [N ] denotes
{1, . . . , N }. The ideas behind results on pattern in primes is that apart from the local obstructions, primes should behave like a random set.
Proposition 3 (Relative Szemerédi’s Theorem). Suppose 0 ≤ f ≤ ν satisfies some pseudorandom
conditions then the statement of the Szemerédi’s Theorem holds i.e.
X
1
f (x)f (x + t)f (x + 2t) . . . f (x + (k − 1)t) ≥ c(α, f ) > 0
(5)
N d+1
d
x∈[N ] ,t∈[N ]
Here, ν is called enveloping sieve constructed via upper bound sieve relying on the work of
Goldston-Yildirim [17]. Basically, it is supported on the set of almost primes which (by sieve
theory) behaves like a random set in many ways and primes is dense in this set. We can think of
ν as the weight that is used to count prime points. It satisfies pseudorandomness conditions called
linear forms conditions and correlation conditions which we don’t define here. The linear forms
conditions are about systems of pairwise linearly independent linear forms and the correlation conditions are about some more general systems of linear forms with possible correlations. Also, it
is important that the dual functions Df which Green-Tao used as the structure part in the decomposition are bounded. In a more recent development, a newer proof of Proposition 3 is given by
Conlon-Fox-Zhao in [7] relying on densification trick that can overcome the correlation condition
and bounded dual conditions. Colon-Fox-Zhao only needs simpler linear form conditions in the
proof. This has applications in more quantitative work, e.g. in the work on prime progressions
with narrow gap by Tao-Ziegler [50] and may relate to our works in session 3.1, 3.2.
3.1. Corners in dense subset of P d and transference principle. Transference principle basically
says that if 0 ≤ f ≤ ν and kν − 1kU k is small then f should behave like functions bounded by 1
when we estimate the expression like (5). Hence, we deduce results for f ≤ ν assuming results
for f ≤ 1 as a black box (where we could exploit the quantitative bound from there as well). This
is first used in Green-Tao [22] and simplified by Gowers in [13] .
We wish to prove an analogue theorem for prime configurations in higher dimensions with
primes replaced by prime points (each coordinate is prime). In higher dimensions, there is a
correlation due to direct product structure. For example, three vertices of a rectangle in Z2 are
prime points then the remaining vertices must be prime. This made it hard to use pseudorandom
strategy to attack the problem. Motivated by the hypergraph approach to the analogue theorem in
integer case, we tried moving our set up to (prime-weighted) hypergraph to reduce the correlation.
It seems that hypergraph setting may be more natural in higher dimensions especially in the
case of prime or relative Szemerédi’s theorem [7], [8] where weaker pseudorandom conditions are
needed in dimension 1. However, for the more quantitative problems, it may be harder to create a
set up on graphs.
We use the hypergraph setting with prime weight ν attached. Surprisingly in the case of corner
x + tF ; F = {(0, 0), (1, 0), (0, 1)} where we have many correlations, our weight setting becomes
very simple and natural. In a joint work with Magyar, we use the transference principle of Gowers
[13] (or Green-Tao [22]) and work on regular hypergraph setting similar to integer case to prove
the following
Theorem 4 (Magyar-T. [35]). Let A ⊂ (PN )d with positive relative upper density α > 0 then A
N d+1
corners for some (computable) constant C(α).
contains at least C(α) (log
N )2d
4
The main difficulty is that Df may not be bounded due to the correlation between the elements
and we have to modify the transference principle accordingly and with this regard, the correlation condition is used to find sets such that Df are bounded. However, due to the application of
hypergraph removal lemma, only linear form conditions are necessary for the remaining parts of
the proof. Our use of correlation condition of Green-Tao in this setting is quite different than the
original proof of Green-Tao’s Theorem. but is more similar to correlation condition on Z[i] in the
work of Tao on Gaussian primes.
In this regards, densification trick of [7] might be able to adapt in this regard to give a simpler
proof where correlation condition of Green-Tao may be omitted. Simpler pseudorandom condition
with transference principle could lead to a more refined quantitative theorem. For example, this
was used in the work of Tao-Zeigler of prime progressions with narrow gap [50]. Since our proof
uses the original weight ν in [21], we have t ≤ N . Narrow corners with small t with the bound
of shape (log N )LF as in [50] might be obtained in this case though it may be hard to transfer this
problem to graph setting. We may try to obtain this for P 2 first following the work of Shkredov
[43].
We see that transference principle could be adapted quite efficiently in some problems with good
bound compared to integer case. We are currently trying to improve the bounds on the number of
prime corners in Z2 to the bound of exponent type similar to the work of Shkredov [43] (which
gives an exponential bound on number of corners in Z2 ) by transferring Shkredov’s idea to the
weighted setting. In [31] Helfgott-Roton used this transference principle to improve the bound in
Roth’s Theorem in the primes of Green [19]. We may combine Shkredov’s idea with transference
principle of Helfgott-Roton [31] which relies on restriction estimate of the enveloping sieve given
by Green and Tao [24].
3.2. Multidimensional Szemerédi’s Theorem in the Primes. In a joint work with Magyar and
Cook, we proved the following theorem using the more general hypergraph method with the weight
ν; this includes the corner problem as a special case. This was also proved in [49] and [33] by TaoZiegler and Fox-Zhao relying on a new weight ν 0 which satisfies stronger (infinite) linear form
conditions relying on Gowers inverse norm theorem [28] and Theorem 5.1 in [23] which currently
gives no explicit bound. It turned out that unlike the integer case, we cannot deduce this general
theorem from the corner case. Our weighted hypergraphs are then no longer uniform and can have
some intermediate weight involving linear forms of 1 < k < d variables. This obstacle prohibits
us from using the transference principle method.
Theorem 5 (Cook-Magyar-T. [8]). Let α > 0 and let ∆ ⊆ Zd be a d-dimensional simplex. There
exists a constant c(α, ∆) > 0 such that for any N > 1 and any set A ⊆ (PN )d such that |A| ≥
α |PN |d , the set A contains at least c(α, ∆)N d+1 (log N )−l(∆) affine copies x + t∆ of the simplex
∆.
In this case, we have to prove a more general version of hypergraph removal lemma where
the hypergraph may not be regular with some possible intermediate weights attached. The main
new ingredient is to prove a parametric version of the regularity lemma (Proposition 2) where we
consider not a single measure system on hypergraph but a family of measure systems depending
on a set of parameters. This allows us to run L2 −energy increment in this setting (and we also
don’t need the correlation condition here). We show that there are many measure systems on a
hypergraph that could be regularized but to a new system of measure that is an extension (in the
sense that the set of parameters involved contains the original set of parameters). This is a kind
5
averaging argument where we only know that many of measures in this family of measure are well
behaved. Our results give a quantitative bound of tower type as it relies on regularity lemma and
give the bound t . N .
The way we analyzed is to use linear form conditions by only consider the set of variables
that our linear forms could depend on to ensure linearly independence. In our proof, the number of
linear form conditions used is large and could depend on α. This is shown to be unnecessary in one
dimension case with simpler linear form conditions in [7] so a more refined analysis is possible.
The set of all parameters involved come from the definition of uniformity norms so we may be
able to see some patterns and do more sophisticated analysis, at least in some special cases. For
example, the recent work of Tao and Ziegler [51] on concatenation theory of Gowers norm (say
in Z2 ) roughly says that if f (m, n) is antiuniform of degree d1 in variable n (one direction) and
antiuniform of degree d2 in variable m (another direction), then f (m, n) are uniform of degree
d1 + d2 − 1. This allows them to get asymptote of some simple nonlinear pattern among primes.
Simpler linear form conditions like in [7] could be helpful along this line.
Another interesting problem is to prove an analogue of Theorem 5 in the case of polynomials.
This would go back to the Theorem of Bergelson-Leibman [1] in the integer case where we would
like to see our method or the ergodic method in this context. This has been done in dimension 1
by Tao-Ziegler [48].
3.3. Circle Method. Circle methods are used to study linear patterns along set of integers using
linear phase functions. Birch and Schmidt [5], [42] use circle method to study the solution sets
of integral forms with some rank conditions (some rank conditions are essential). Some ideas
from additive combinatorics are used along this line where solutions are restricted to some special
sets like primes or almost primes. For example, Bourgain-Gamburd-Sarnak’s result [6] on almost
primes uses the idea of affine sieves. Cook-Magyar’s result [9] on prime solutions utilizes ideas of
regularization (with tower-type bound on rank condition).
In [36] Magyar-T proved an analogue result of Birch under the condition of that each coordinate
in each solution has all prime factors are bigger than 1 . The idea is that almost prime should
behave randomly like integers so we applied Goldston-Yildirim sieve [17]. The only difference is
the requirement of solutions in p-adic unit to have positive singular series.
The first application of inverse Gowers norm theorem is to study the asymptotic of prime solutions in systems of linear equation of finite complexity by Green-Tao [23]. Recently, so-called
nilpotent circle method where nilsequences could play a role of the linear phase, is also used to find
the asymptotic of average of f (L1 (u, v)) . . . f (Lk (u, v)) for some classes of arithmetic functions
f and Lj are binary linear forms. Some arithmetic functions that are orthogonal to polynomial
nilsequences are studied in [37]. In [29] Frantzikinakis and Host prove a structure theorem for
bounded multiplicative functions using higher order Fourier analysis with some new applications
in number theory and Ramsey theory. It may be possible to extend our results in number theory
with these tools. For example, giving asymptote of some quadratic pattern among primes or almost
primes, or proving stronger pseudorandomness conditions.
4. E RGODIC R AMSEY T HEORY
4.1. Triangle and Squares. The following theorem first proved by Furstenberg-Katnelzon-Weiss
[10] and later generalized by Ziegler [54].
6
Proposition 6 (Ziegler). Suppose E ⊆ Rd has positive upper density and F = {0, v1 , . . . , vd } ⊆
Rd . Let Eδ denotes the δ−neighborhood of E. Then Eδ contains an isometric copy l · F of F for
all l ≥ l(F, δ).
In the discrete case, Graham [19] provided a counterexample to the case when F is not contained
in a sphere. Therefore, we have the following conjecture.
Conjecture 1. Suppose E ⊆ Zd has positive upper density, let F = {0, v1 . . . , vm } be such that
v1 , . . . , vk are on the same sphere. Then assuming the dimension d ≥ d(E, F ) is sufficiently large
then there is an integer q such that for all l ≥ l(E) which is a multiple of q, E contains isometric
copy l · F
This conjecture is solved in [34] in the case that {v1 , . . . , vm } are linearly independent. The
method is based on asymptotic expressions of the Fourier transform of lattice points on certain
homogeneous varieties which may not be easy to generalize. This is related to diophantine equation
as in Section 3.3, for example, the case F = {0, v1 } would correspond to distance problem; solving
x21 + . . . x2d = l. In this case, the circle method works well only when d ≥ 5.
An interesting case not covered by the above result which I considered along with Ákos Magyar is when we have square configurations where vectors are not linearly independent. This is
equivalent to studying the following multiple recurrence
X
1 X
f (x)f (T m1 x)f (T m2 x)f (T m1 +m2 x).
SN
|m1 |=N |m2 |=N
m1 ·m2 =0
Here SN is the normalizing factor. We need to identify the correct characteristic factor. However
due to the quadratic correlation of the index in the average: k 2 − (k + m1 )2 − (k + m2 )2 + (k +
m1 + m2 )2 = 0, we may expect higher order characteristic factor and nilsequence may play a role
in studying this average. Also, the condition m1 · m2 = 0 makes this much more difficult and the
usual techniques may not be applied directly. It may be a good idea to come up with a new proof
of results in [34] first, using ideas of characteristic factor which is supposed to be the Kronecker’s
Factor corresponding to classical Fourier analysis.
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E-mail address: tatchai@math.ubc.ca
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