Several results in this chapter relied on facts concerning the distribution of the primes
P = {2,3,5, . . .}.
The distribution of this set is of course a very well-studied subject in analytic num- ber theory, with one of the fundamental results being theprime number theorem
|P∩[1,n]| =(1+o(1)) n
logn. (1.44)
An equivalent formulation is that if pk denotes the kth prime, then pk= (1+o(1))klogk. The famous Riemann hypothesis, which is still unsolved, is equivalent to the stronger statement that
|P∩[1,n]| = n
2
d x
logx +Oε n1/2+ε
(1.45) for anyε >0, or equivalently that pk=klogk+Oε(k1/2+ε) for anyε >0.
The prime number theorem is rather deep and will not be proven here. In this Appendix we present some related results, most of which have surprisingly elemen- tary and beautiful proofs. As they are number-theoretical rather than probabilistic in nature we have chosen to place these results in an appendix to this chapter.
We begin with some classical estimates of Chebyshev and Mertens). As is customary, when summing over a variable p,pis understood to denote a prime.
Proposition 1.51 (Elementary prime number estimates) Let n≥1be an inte- ger. Then we have the estimates
p≤n
logp =O(n) (1.46)
p≤n
logp
p =logn+O(1) (1.47)
p≤n
1
p =log logn+O(1). (1.48)
Remark 1.52 With the prime number theorem, we can improve (1.46) to
p≤nlogp=(1+o(1))n, but it is not necessary to do so for our applications here.
Proof We first prove (1.46). Without loss of generality we may taken to be a power of two. Consider the binomial2n
n
. From Pascal’s formula we know that 2n
n
≤4n. On the other hand, it is clear that every prime betweenn and 2n will divide2n
n
. Thus
n<p≤2n
p≤4n. Taking logarithms we conclude
n<p≤2n
logp=O(n).
Applying this bound ton/2,n/4, and so forth, and then summing the geometric series, the claim (1.46) follows.
Now we prove (1.47). This is a similar argument but based around the factorial n! instead of2n
n
. Observe that the only primes dividingn! are those less than or equal ton. For each prime p≤n, there aren/pnumbers (between 1 andn) divisible byp,n/p2numbers (between 1 andn) divisible byp2and so on. Thus
n!=
p≤n
pn/p+n/p2+ããã. (1.49) Taking the logarithm of both sides and applying Stirling’s formula (Exercise 1.10.1) we obtain
nlogn+O(n)=
p≤n
(n/p + n/p2 + ã ã ã) logp.
Since
n/p + n/p2 + ã ã ã = n
p +O(1)+O n
p2 ,
we conclude, after some rearranging, that
p≤n
nlogp
p =nlogn+O(n)+
p≤n
O(logp)+
p≤n
O
nlogp p2 . Since
k logk
k2 is convergent, the last term isO(n). The claim now follows from (1.46).
We shall deduce (1.48) from (1.47) using Abel’s summation technique, rewriting one partial sum over primes as an average of others. Observe from the fundamental theorem of calculus that
1
p =logp p
1 logp
=logp p
∞
1
I(t >p) dt tlog2t and hence
p≤n
1
p =
p≤n
logp p
∞
1
I(t> p) dt tlog2t. Swapping the sum and integral, we obtain
p≤n
1 p =
∞
1
p≤t
logp p
dt tlog2t. Applying (1.47), we obtain
p≤n
1 p =
∞
1
(logt+O(1)) dt tlog2t.
Since logtlogt2tis the antiderivative of log logt, andtlog12tis absolutely convergent,
the claim follows.
We now turn to a deeper fact concerning the distribution of primes in intervals.
Theorem 1.53 For all sufficiently large n, we have|P∩[n−x,n)| =(logxn) for all n2/3 <x<n.
Results of this type first appeared by Hoheisel [183]; the result as claimed is due to Ingham [188]. Note that this theorem follows immediately from the Riemann hypothesis (1.45). However, this theorem can be proven without using the Riemann hypothesis, rather some weaker (but still very non-trivial) facts on the distribution of zeroes of the Riemann zeta function: see [170]. We remark that if one only seeks the upper bound on|P∩[n−x,n)|then one can use relatively elementary sieve theory methods to establish the claim. The constant 2/3 has been lowered
(the current record is 7/12, see [187], [178]). However, for the applications here, any exponent less than 1 will suffice.
We now combine this theorem with the Abel summation method to establish some further estimates on sums involving primes.
Proposition 1.54 Let n be a large integer. Then we have the estimates
p∈P∩[1,n−n2/3)
1
n−p =(1) (1.50)
p∈P∩[1,n−n2/3)
log(n−p)
n−p =(logn). (1.51)
Proof We begin by proving (1.50). From the fundamental theorem of calculus we have
1 n−p =
∞
1
1p∈[n−x,n−n2/3)
1 x2 d x for all p∈ P∩[1,n−n2/3), and hence
p∈P∩[1,n−n2/3)
1 n−p =
∞
1
P∩
n−x,n−n2/3d x x2.
The integrand vanishes whenx≤n2/3. Whenn2/3<x≤2n2/3, Theorem 1.53 shows that the integrand isO(n2/31logn), while forx≥n2/3another application of Theorem 1.53 shows that the integrand is (xlog1 n) whenx≤n and(x2logn n) whenx >n. Putting all these estimates together we obtain (1.50). The estimate (1.51) then follows immediately from (1.50) since log(n−p)=(logn) when
p∈[1,n−n2/3].
Exercises
1.10.1 By approximating the sumn
m=1logmby the integraln
1 logx d x, prove Stirling’s formula
logn!=nlogn−n+O(logn) (1.52) for alln>1.
1.10.2 Using Proposition 1.51, show that there is a constantcso that there is always a prime betweennandcnfor every positive integern.
1.10.3 By being more careful in the proof of (1.46), show that
p<n
logp ≤2nlog 2+O n1/2
and
n≤p<2n
logp+
p≤2n/3
logp≥2nlog 2−O n1/2
,
and concludeBertrand’s postulate, namely that for every sufficiently large integern there exists a prime betweennand 2n. (This argument is due to Ramanujan. Bertrand’s postulate in fact holds for all integersn, as the case of smallncan be verified directly.)
1.10.4 Without using the prime number theorem, prove that |P∩[1,n]| = (lognn); this is known asChebyshev’s theorem. This theorem is of course superseded by the prime number theoremπ(n)=(1+o(1))lognn, but has the advantage of having a short elementary proof.
1.10.5 Prove thatpk=(klogk), where pkdenotes thekth prime. Again, this is superseded by the prime number theorem pk=(1+o(1))klogk.
1.10.6 Define thevon Mangoldt function:Z+→Rby setting(n) :=logp ifn>1 is a power of a prime p, and(n)=0 otherwise. Show that
d|n
(d)=logn (1.53)
for all integersn≥1. Use this to prove that ∞
n=1
(n) ns
∞ n=1
1 ns
=∞
n=1
logn ns
for all real numberss>1. Also, use (1.53) to give an alternative proof of (1.49).
1.10.7 Using the preceding exercise, show that ∞
n=1
logp ps = 1
s−1 +O(1) for alls>1; integrate this to conclude
∞ n=1
1
ps =log 1
s−1+O(1) (1.54)
for alls>1. Show that these estimates can also be deduced from Propo- sition 1.51 via Abel’s method. Conversely, use (1.54) and (1.46) to give an alternative proof of (1.48).
1.10.8 Using Abel’s summation method, show that the prime number theo- rem π(x)=(1+o(1))logxx is equivalent to the estimate
n≤x(n)= (1+o(1))x.
1.10.9 By being more careful in the proof of (1.48), show that
p<n
1
p =log logn+C+O 1
logn
for some absolute constantC. Use this to deduceMerten’s theorem
p<n
1− 1
p =(1+o(1)) C
logn (1.55)
for some other absolute constantCand alln >1. (In fact one hasC= e−γ, whereγ =0.577. . .is Euler’s constant.)
Sum set estimates
Many classical problems in additive number theory revolve around the study of sum sets forspecificsetsA,B(though one typically works with infinite sets rather than finite ones). For instance, ifN∧2 := {0,1,4,9,16, . . .}is the set of square numbers, then it is a famous theorem of Lagrange that 4N∧2=N, i.e. every natural number is the sum of four squares; if P:= {2,3,5,7,11, . . .}is the set of prime numbers, then it is a famous theorem of Vinogradov that (2ãN+1)\3P is finite (i.e. every sufficiently large odd number is the sum of three primes); in fact it is conjectured that this exceptional set consists only of 1, 3, and 5. The corresponding result for (2ãN)\2Premains open; the infamousGoldbach conjectureasserts that 2Pcontains every even integer greater than 2, but this conjecture remains far from resolution.
In this text, we shall not focus on these types of problems, which rely heavily on the specific number-theoretic structure of the sets involved. Instead, we shall focus instead on the analysis of sum sets A+B and related objects for more general sets A,B. To simplify the discussion we shall focus primarily on additive sets A,B, which are finite and non-empty subsets of an additive group such asZ; thus our theory will not cover infinite sets such as the squares N∧2 or the primes P directly, although one can certainly use this theory to analyze those sets simply by considering finite truncations, say to an interval [0,N].
A fundamental problem in this field is theinverse sum set problem: ifA+Bor A−Bis small, what can one say aboutAandB? A more specific question is as follows: ifAis a finite non-empty subset of integers such that|A+A| =K|A|for some small numberK, what can one say aboutA? Here and in the rest of the text we use|A|to denote the cardinality of a finite setA. The numberK := |A+A|/|A|is referred to as thedoubling constantofAand will be denoted in this text byσ[A]. It is easy to see that this constant is at least 1, but it can be much larger; for instance, if A is a geometric progression such as A=2∧[0,N)= {1,2,22, . . . ,2N−1}
51
then one can easily verify that σ[A]=(N +1)/2, so the doubling constant can be arbitrarily large; indeed for “generic” sparse sets Awe will haveσ[A]= (|A| +1)/2.
At the other extreme, if Ais an arithmetic progression A:=a+[0,N)ãr = {a,a+r,a+2r, . . . ,a+(N−1)r}of length N then one can check that Ahas doubling constantσ[A]=2− N1. Thus arithmetic progressions are examples of sets with small doubling constant. One can perturb this example to produce a number of other examples of sets with small doubling constant; for instance ifAis the above arithmetic progression, and we letAbe a subset ofAof cardinalityN/2 (say), then one can easily check thatAhas doubling constant at most 4. Another example comes from adding an arbitrary integerntoA; then the setA∪ {n}also has doubling constant at most 4.
One can generalize the concept of an arithmetic progression, to create more sets with small doubling constant. Consider the set
A:=a+[0,(N1,N2))ã(v1, v2)={a+n1v1+n2v2: 0≤n1<N1; 0≤n2<N2}, wherea, v1, v2are integers,N1,N2are positive integers andn1,n2are understood to lie in the integers; this is an example of ageneralized arithmetic progression of rank2. One can verify that such sets have a doubling constant of at most 4. Note that such sets can look quite different from an ordinary arithmetic progression if N1,N2are large andv1, v2are very widely separated.
We have just remarked that generalized arithmetic progressions have small doubling constant. One of the fundamental theorems in this subject isFreiman’s theorem, which asserts a partial converse to this claim. Freiman’s theorem shows that any finite subset of the integers with small doubling constant can be efficiently contained in a generalized arithmetic progression (of bounded rank). This theorem is very useful, but is rather deep, and we will defer its proof to Section 5.4. It also has the drawback that some of the constants in this theorem depend exponentially on the doubling constantσ[A]. As such, it tends to only be useful in contexts where the doubling constantσ[A] is of the order of log|A|or smaller.
Roughly speaking, one can classify results in inverse sum set theory by the range ofσ[A] for which the results are non-trivial. The caseσ[A]=1 is group theory (see Proposition 2.7). Whenσ[A] is very small, e.g.σ[A]<2 orσ[A]<3, we have a complete characterization of the inverse problem, characterizingAin terms of groups and arithmetic progressions (see Corollary 5.6, Theorem 5.11). When σ[A]=O(log|A|), the best result is Freiman’s theorem, which characterizes A in terms of generalized arithmetic progressions. Whenσ[A]=O(|A|ε) for some small ε, we have Proposition 2.26 (as well as many of the other results in this chapter), which characterizesAin terms of approximate groups. In the remaining
cases |A|εσ[A]≤ |A|, some of the estimates here are still useful, but our understanding is still quite poor.
We will not prove Freiman’s theorem in this chapter. However, we will develop the more elementary theory of sum set estimates, which can be used as sub- stitutes for Freiman’s theorem in some cases and are also of interest in their own right; this theory will also be needed in the proof of Freiman’s theorem later on. These estimates are obtained by very simple combinatorial considera- tions, and rely on simple arithmetic facts such asa−c=(a−b)+(b−c) and a+b=a+b ⇐⇒ a−b=a−b. Because of the simplicity of the tech- niques used here, the results in this section are quite general, being applica- ble to any additive group and even to a large extent to non-abelian groups (see Section 2.7); we will wait until Chapter 5 until developing sum set estimates which exploit the specific structure of the ambient group (though see also Section 3.4).
Also, the bounds obtained here are fairly reasonable, for instance the dependence of constants on the doubling constantσ[A] is only polynomial in all the results in this section (in contrast to the exponential dependence onσ[A] in Freiman’s theorem). In some cases, though, the results in this section will be superseded by more precise results proven using advanced techniques, which we will address in later sections; for instance, in Section 6.5 we shall develop the theory ofPl¨unnecke inequalities, which give more precise control on iterated sum sets and also han- dle the case when AandBhave very different sizes, a case which is not treated efficiently by the tools in this section.
There are a large number of results in this chapter, but we point out a couple of specific results proven here which have a very large number of applications.
The first isRuzsa’s triangle inequality, Lemma 2.6, which allows us to define a
“metric” on the space of additive sets and which measures how small their sum sets are. Then there is Corollary 2.12, which links the size of|A+B|and|A−B|
for arbitrary additive sets A,B. This generalizes to the iterated sum set estimates in Corollary 2.23 and Corollary 2.24. Another very useful class of tools are the covering lemmas – Ruzsa’s covering lemma (Lemma 2.14), Green–Ruzsa’s cov- ering lemma (Lemma 2.17), and Chang’s covering lemma (Lemma 5.31), which gives conditions under which one set Acan be efficiently covered by translates of another set B. These results are collected together in Proposition 2.26 and Proposition 2.27, which characterize sets with small sum set in terms ofapprox- imate groups. Last, but certainly not least, there is theBalog–Szemer´edi–Gowers theorem, which generalizes the previous results to the setting when one has only partial information on a sum set (or equivalently, one only controls the “additive energy” between two sets); see Theorem 2.29 and Theorem 2.31. We also develop an asymmetric version of this theorem in Section 2.6.