Corollary 3.9 Fundamental theorem of finitely generated additive groups)
8.5 The sum-product problem in other fields
A natural extension of the sum-product problem is to consider sets from fields and rings other thanR. One example (whenRis replaced byZp for a prime p) was consider in an earlier chapter. In this section, we consider the case whenRis replaced by the set of complex numbers.
One way to attack the problem is to prove a complex version of Szemer´edi–
Trotter theorem and then repeat the proofs of Theorems 8.14 and 8.15. While it is believed that the statement of Szemer´edi–Trotter theorem holds for complex lines and points, proving it is not easy as the technique using the crossing number no longer applies (see however the recent announcement by T´oth [368]).
In the following, we show that using a clever double counting argument, one can extend Elekes’s result for complex numbers. In fact, the argument, which is
due to Solymosi [325], is effective for several other number fields as well. (See the remark at the end of the proof.)
Theorem 8.24 [325] For any finite non-empty sets of complex numbers A,B, and Q,
|A+B| ã |AãQ| =
|A|3/2|B|1/2|Q|1/2 . By settingQ=B=A, it follows immediately that
|A+A| ã |AãA| =
|A|5/2 and
|A+A| + |AãA| =
|A|5/4 , thus this theorem generalizes Theorem 8.14.
Proof We may assume|A| ≥2 and 0=Q. From elementary algebra we observe that the map
(a,a,b,q)→(a+b,a+b,aq,aq)
is one-to-one from AìAìBìQto (A+B)ì(A+B)ì(AãQ)ì(AãQ) provided that we exclude the diagonala=a. This observation by itself is only enough to obtain the trivial bound|A+B| ã |AãQ| =(|A||B|1/2|Q|1/2). How- ever we can do better by exploiting the intuitive observation that ifais close to a, thena+bis close toa+bandaqis close toaq.
More precisely, for eacha∈ A, define thenearest neighbor a ofa to be an element of A\a which minimizes the distance|a−a|. (If there is more than one candidate for nearest neighbor, choose arbitrarily.) We refer to (a,a) as a neighboring pair, thus there are|A|neighboring pairs. We caution that if (a,a) is a neighboring pair then (a,a) is not necessarily a neighboring pair also.
Call a quadruple (a,a,b,q)goodif (a,a) is a neighboring pair,b∈ B and q ∈Q, and one has the closeness properties
|{u ∈ A+B:|a+b−u| ≤ |a−a|}| ≤ 28|A+B|
|A| (8.7)
and
|{v∈ AãQ:|aq−v| ≤ |aq−aq|}| ≤ 28|AãQ|
|A| . (8.8)
Informally, (8.7) and (8.8) assert thata+bis a fairly close neighbor ofa+bin A+B, and similarlyaq is a fairly close neighbor ofaqinAãQ. We will apply a double counting argument toN, the number of good quadruples.
First we establish a lower bound. For eacha∈ AletDa:= {z∈C:|z−a| ≤
|a−a|}be the disk of radius|a−a|centered ata. A simple geometric argument (which we leave as an exercise) shows that any complex numberzcan be contained in at most seven of these disks. In particular for anyb∈ Bwe have
a∈A
|{u∈A+B:|a+b−u| ≤ |a−a|}| =
z∈A+B−b
|{a∈A:z∈Da}| ≤7|A+B|
and similarly for anyq ∈Q
a∈A
|{v∈AãQ:|aq−v| ≤ |aq−aq|}| =
z∈AãQ/q
|{a∈A:z∈Da}| ≤7|AãQ|.
If we thus fixbandqand choosea∈ Auniformly at random, a simple application of Markov’s inequality then shows that (a,a,b,q) will be good with probability at least 1/2. This shows that
N ≥ |B||Q||A|
2 .
Now we establish an upper bound. Recall that the quadruple (a,a,b,q) is uniquely determined by the quadruple (a+b,a+b,aq,aq). There are|A+B|choices fora+band|AãQ|choices foraq. For fixeda+b, we see from (8.7) that there are at most 28||AA+|B|elements ofA+Bwhich are closer to or equally distant from a+bthana+b, and thus there are at most 28|A+B||A| values ofa+b. Similarly there are at most 28||AAã|Q| values ofaq. This gives the upper bound
N ≤ |A+B|28|A+B|
|A| |AãQ|28|AãQ|
|A| .
Combining this with the lower bound, we obtain the claim.
Remark 8.25 A similar argument works for quaternions and for other hypercom- plex numbers. In general, ifTandQare sets of similarity transformations andAis a set of points in space such that, from any quadruple (t(p1),t(p2),q(p1),q(p2)), the elements t ∈T, q ∈Q, and p1= p2∈ A are uniquely determined, then c|A|3/2|T|1/2|Q|1/2≤ |T(A)| ã |Q(A)|,wherecdepends on the dimension of the space only.
To conclude this section, let us describe a recent result of Chang, who investi- gates the sum-product problem for matrices [51].
Theorem 8.26 There is a function(n)tending to infinity with n such that the following holds. Let d be a fixed integer and A be a finite set of d×d real matrices such that for any two different elements M and Mof A,det(M−M)=0. Then
|A+A| + |AãA| ≥(|A|)|A|.
Theorem 8.27 For every d there is a positive constant =(d) such that the following holds. Let A be a finite set of d×d real, symmetric, matrices. Then
|A+A| + |AãA| ≥ |A|1+.
The proofs of these theorems are more complicated than those presented here and we refer the readers to [51] for details.
Exercise
8.5.1 With the notation in the proof of Theorem 8.24, show that every complex number is contained in at most seven of the disksDa. (Hint: show that if zis contained in bothDaandDawitha,a,zdistinct, thena,asubtend an angle of at least 60◦with respect toz.)
Algebraic methods
In most of this book we have studied additive combinatorics problems in an ambient groupZ, relying primarily on the additive structure ofZ(as manifested for instance in the Fourier transform). However, in many cases the ambient group is in fact afield F, and thus supports a number of special functions, in particular polynomials. One can then use tools from algebraic geometry to exploit these polynomial structures;
this is known as thepolynomial method. One of the primary ideas here is to interpret an additive set (e.g. a sum setA+B) as the zero locus of one or more polynomials, possibly in several variables. One can then hope to control the size of such sets using results from algebraic geometry about the number and distribution of zeroes of polynomials. The most familiar example of such a theorem is the statement that a polynomial P(t) of one variable with degree d in a field F can have at most d zeroes; however for most applications we will need to study the zero locus of polynomial(s) in many variables. In this chapter we present four related tools and techniques from algebraic geometry which allow one to control such a zero locus.
The first is the powerful combinatorial Nullstellensatz of Alon (Theorem 9.2), which asserts that the zero locus of a polynomial P(t1, . . . ,tk) cannot contain a large boxS1ì ã ã ã ìSkif a certain monomial coefficient ofPis non-vanishing; this is particularly useful for obtaining lower bounds on the size of restricted sum sets and similar objects. The second is theChevalley–Warning theorem(Theorem 9.24), which shows that under certain conditions the cardinality of a zero locus of multiple polynomials must be a multiple of char(F), the characteristic of the underlying field. This is useful for demonstrating the existence of non-trivial solutions to a set of polynomial equations inF. The third isStepanov’s method(see Section 9.7), which obtains upper bounds on a set by using linear algebra methods to locate a polynomial that vanishes to very high order at each of the elements of the set;
this has proven to be particularly useful for controlling additive combinations of multiplicative subgroups of a finite field, and thus has application to sum-product estimates. Finally we discussdivisibility criteria, which show that a polynomial
329
cannot have certain types of zeroes if some combination of its coefficients are divisible (or not divisible) bypin a certain manner; the most well known example of this is Eisenstein’s criterion (Exercise 9.8.2), but the combinatorial Nullstellensatz can also be viewed as a statement of this type, and another example arises in cyclotomic fields (Lemma 9.49). As an application of these criteria we present an uncertainty principle forZpwhich gives a Fourier-analytic proof of the Cauchy–
Davenport inequality (Theorem 5.4).
Much of the theory pertains to arbitrary fields F. However, we will at times need to focus on two special types of fields. The first arefinite fields, of which the primary example are the fieldsFp=Zpof prime order. We shall review the theory of these fields in Section 9.4. The second are thecyclotomic fields, generated by
pth roots of unity; we shall review the theory of those fields in Section 9.8.
It is easy to see that in a fieldF, all non-zero elements have the same torsion as the identity element 1. We refer to this torsion as thecharacteristicchar(F) ofF;
it is either zero (ifF is torsion-free) or a prime p(which is for instance the case when Fis finite). Some of our results will only hold if the characteristic of Fis sufficiently large (or equal to zero).