NEIL LYALL A N. log log N

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1 SÁRKÖZY S THEOREM NEIL LYALL. Introduction The purpose of this expository note is to give a self contained proof (modulo the Weyl ineuality) of the following result, due independently to Sárközy [5] and Furstenberg []. Theorem. Let δ >, then there exists N = N (δ) such that if N N and A [, N] with A δn, then A necessarily contains distinct elements a, b whose difference a b is a perfect suare. We will closely follow the approach taken in Lyall and Magyar [3] and employ a density increment strategy to obtain the result above with () N = exp(cδ log δ ). This is euivalent to the statement that if A [, N] and d / A A for any d, then necessarily () A N C log log N log N.. Dichotomy between randomness and arithmetic structure Our approach will be to obtain a dichotomy between randomness and structure of the following form. Proposition. Let A [, N] and δ = A /N. If N δ C with C sufficiently large, then either N ( (3) A A + d ) / δ A N d= or there exists a suare difference arithmetic progression P in [, N] of length P cδ 5 N such that (4) A P > δ( + cδ) P. Proof that Proposition implies Theorem (with bound ()). Suppose A [, N], with A δn, contains no suare differences, then Proposition allows us to perform an iteration. At the kth step of this iteration we will have a set A k [, N k ] of size δ k N k. This set will be an appropriately rescaled version of a subset of A itself and hence will also contain no suare differences. Let A = A, N = N and δ = A /N δ. Proposition ensures that either (5) N k δ C k or else the iteration proceeds allowing us to choose N k+, δ k+ and A k+ such that and N k+ cδ 5 kn k δ k+ δ k + cδ k. Now as long as the iteration continues we must have δ k, and so after O(δ ) iterations condition (5) must be satisfied, giving δ Cδ N δ C log N Cδ log δ. The rest of this note is devoted to the proof of Proposition. As opposed to the standard L increment strategy of Roth, we will obtain the dichotomy in Proposition by exploiting the concentration of the L mass of the Fourier transform (this sometimes referred to as an energy increment strategy).

2 NEIL LYALL 3. Setting the stage for the proof of Proposition Let A [, N], with N δ C, where δ = A /N and C is a sufficiently large constant. Our approach will be to assume that A exhibits neither of the two properties described in Proposition and then seek a contradiction. 3.. A simple conseuence of A being non-random. If we were to suppose that A is non-random, in the sense that ineuality (3) does not hold, then it would immediately follows that (6) A (m) A (n) S (m n) 4 δ A S where m,n Z (7) S = d : d N/9}. 3.. A simple conseuences of A being non-structured. If we were to assume that A is regular, in the sense that A in fact satisfies the ineuality for all arithmetic progressions P [, N] of the form A P δ( + ε) P (8) P = m + l : l L} with L δ 4 εn, then the set A (N/9, 8N/9] must contain most of the elements of A. In particular (9) A (N/9, 8N/9] (3/4) A, since if this were not the case we would immediately obtain a progression P [, N] of the form (8) with = and L N/9 such that A P δ( + /8) P The balance function. We define the balance function of A to be () f A = A δ [,N]. We note that f A has mean value zero, that is f A (m) =, this will be critically important later. It easy to verify that if A satisfies ineualities (6) and (9), then () f A (m)f A (n) S (m n) 4 δ B S. m,n Z One can see this by simply expanding the sum into a sum of four sums, one involving only the function A on which we can apply (6), two involving the functions A and δ [,N] on which we can apply (9), and one involving only the function δ [,N] which can be estimated trivially Fourier analysis on Z. If f : Z C and has finite support, then we define its Fourier transform by f(α) = m Z f(m)e πimα. The finite support assumption on f ensures that f is a continuous function on the circle, and in this setting the Fourier inversion formula and Plancherel s identity, namely f(m) = f(α)e πimα dα and f(α) dα = m Z are simply immediate conseuences of the familiar orthogonality relation e πimα if m = dα =. if m f(m) It is then easy to verify that from ineuality () we immediately obtain the estimate () f A (α) S (α) dα 4 δ B S

3 SÁRKÖZY S THEOREM 3 where we recognize (3) S (α) = as a classical Weyl sum. N/9 d= e πidα, 3.5. Estimates for Weyl sums. Let M = N/9. We note that whenever α M there can be no cancellation in the Weyl sum (3), in fact the same is also true when α is close to a rational with small denominator (i.e. there is no cancellation over sums in residue classes modulo ). We now state a precise formulation of the fact that this is indeed the only obstruction to cancellation. For η > we define (4) M = M (η) = Proposition 3. Let η >. α [, ] : α a η M (i) (Minor arc estimate) If α / M for any η, then S (α) ηm + O(M /4 ). (ii) (Major arc estimate) If α M for some η, then S (α) C / M + O(M / ). } for some a [, ]. The proof of this result is a straightforward (and presumably well known) conseuence of the standard estimates for Weyl sums, for the sake of completeness we include these arguments in an appendix. 4. The proof of Proposition In the previous section we established that ineualities (6) and (9) would be immediate conseuences of A not exhibiting either of the two properties described in Proposition. We now present the two lemmas from which we will obtain our desired contradiction. In both lemmas below we assume that A [, N] and N δ C, where δ = A /N and C is a sufficiently large constant. Moreover, we set η = cδ, where c is a sufficiently small constant. Lemma 4. If A is neither random nor structured, in the sense outlined in Proposition, then there exists η such that (5) f A (α) dα cδ. δ A M The second lemma is a precise uantitative formulation, in our setting, of the standard L density increment lemma. Lemma 5. Let ε η /4π. If A is regular, in the sense that A P δ( + ε) P for all progressions P [, N] of the form (8) with L η εn, then (6) f A (α) dα ε. δ A M We therefore obtain a contradiction if ε cδ, proving Proposition.

4 4 NEIL LYALL 4.. Proof of Lemma 4. It follows from the minor arc estimate of Proposition 3 (since N δ C for C sufficiently large) and Plancherel s identity that f A (α) S (α) dα CηM A. minor arcs Therefore, if η = δ/8c, it follows from estimate () and the major arc estimate of Lemma 3 that It therefore follows that as reuired. M / f A (α) dα η A. η = max f A (α) dα η A η M 4.. Proof of Lemma 5. We fix and L so that L = η εn and define Claim. If α M, then P (α) P /. P = l l L}. Proof of Claim. Since L α L η M = η ε, for all α M, where denotes the distance to the nearest integer, it follows that L P (α) P e πil α ) P ( πl α P /, for all α M, provided ε η /4π. l= We now note that Plancherel s identity (applied to the function f A P ) and Claim imply that (7) f A (α) 4 dα δ A M δ A P f A P (m). The conclusion of Lemma 5 will therefore be an immediate conseuence of the following. m Z Claim. As a conseuence of the assumptions in Lemma 5 if follows that f A P (m) 3ε δ A P. m Z Proof of Claim. We let M = m Z m P [, N]} E = ([, N] + P ) \ M and write f A P (m) = f A P (m) + f A P (m). m Z m E We note that since f A P (m) = A (m P ) δ [, N] (m P ) it follows from our regularity assumption on A that if m M, then while for m E we can only conclude that Now since f A has mean value zero the convolution δl f A P (m) δεl, f A P (m) = n f A P (m) L. f A (n) P (m n)

5 SÁRKÖZY S THEOREM 5 also has mean value zero. Thus, using the fact that g = g + g, where g + = maxg, } denotes the positive-part function, and the trivial size estimate M N, we can deduce that ( ) f A P (m) sup f A P (m) (f A P ) + (m) (δl)(δεl) M δ εl N. Using the fact that E L = η εn, it follows that f A P (m) P E m E provided η δ. This concludes the proof of Claim and establishes Lemma 5. f A P (m), Appendix A. Weyl Sum Estimates: Proof of Proposition 3 A.. Standard estimates for Weyl sums. Let S M (α) = M e πidα. Proposition 6 (The Weyl ineuality). If α a/ and (a, ) =, then d= S M (α) M log M(/ + /M + /M ) /. The proof of this result is completely standard, see for example [4] or []. We remark that this gives a non-trivial estimate whenever M µ M µ for some < µ <. Let (8) M a/ α = [, ] : α a }. M / We say that α is in a minor arc if α / M a/ for any (a, ) = with M /. Lemma 7 (Minor arc estimate I). If α / M a/ for any (a, ) = with M /, then (9) S M (α) CM /4. While on the complement of these minor arcs (the major arcs) we have the follow important estimate. Lemma 8 (Major arc estimate). If α M a/ for some (a, ) = with M /, then () S M (α) CM / ( + M α a/ ) / + O(M / ). For the proofs of these two lemmas see Section A.3. A.. Refinement of the major arcs. We now let < η and define () M a/ = α : α a } η M. Lemmas 7 and 8 combine to give the following result from which Proposition 3 is an immediate conseuence. Lemma 9 (Minor arc estimate II). If α / M a/ for any (a, ) = with η, then S M (α) ηm + O(M /4 ). Proof. It follow from Lemma 8 that on M a/ we have S M (α) ηm provided (a, ) = and either η N / or η M α a/ M +/.

6 6 NEIL LYALL A.3. Proof of Lemmas 7 and 8. Before launching into this, we make the important observation that the major arcs (and hence the refined major arcs) are a union of (necessarily short) pairwise disjoint intervals. Lemma. If a/ a / with, M /, then M a/ M a / =. Proof. Suppose that M a/ M a /. Using the fact that a a, we see that a contradiction. M a / a a a = M, /5 Proof of Lemma 7. It follows from the Dirichlet principle and the fact that α is in a minor arc that there exists a reduced fraction a/ with M / M / such that α a/. It therefore follows from the Weyl ineuality that S M (α) 3M / log M CM /4. Key to the proof of Lemma 8 is the following approximation. Lemma. If α M a/ with M /, then () S M (α) = S(a, )I M (α a/) + O(M /5 ) where S(a, ) := r= e πiar / and I M (β) := M e πiβx dx. Proof. We can write α = a/ + β where β /M / and M /. We can also write each d M uniuely as d = m + r with r and m. It then follows that Since and S M (α) = = r= m= r= e πi(a/+β)(m+r) + O() e πiar / m= e πiβ(m+r) + O(). e πi(m+r)β e πim β e πi(mr+r )β Cdr β CM +/ e πim β m= e πix β dx m= m= m+ m M / e πim β e πix β dx π(m + ) β it follows that SM (α) S(a, )I M (β) CM /5. Lemma 8 then follows almost immediately from the two basic lemmas below. Lemma (Gauss sum estimate). If (a, ) =, then S(a, ). More precisely, if odd S(a, ) = if mod 4. if mod 4

7 SÁRKÖZY S THEOREM 7 Lemma 3 (Oscillatory integral estimate). For any λ e πiλx dx C( + λ) /. Proof of Lemma 8. Lemmas and 3 imply that the main term in () S(a, )I M (α a/) M / ( + M α a/ ) /, and since / M / and M ( α a/ M /, it follows that Proof of Lemma. Suaring-out S(a, ) we obtain M / ( + M α a/ ) / M 9/ M /5. S(a, ) = e πia(r s )/. s= r= Letting r = s + t and using the fact that (a, ) = and e πia(st)/ if at mod = otherwise it follows that S(a, ) = t= s= e πiat / e πia(st)/ = Proof of Lemma 3. We need only consider the case when λ. We write e πiλx dx = s= λ / e πiλx dx + if odd ( e πia(/4) + ) if even. e πiλx λ / dx =: I + I. It is then easy to see that I λ /, while integration by parts gives that I = ( d ) λ 4πiλx dx eπiλx dx / [ ] 4πλ x eπiλx λ x eπiλx dx / Cλ /. λ / + References [] H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. d Analyse Math, 7 (977), pp [] W. T. Gowers, Additive and Combinatorial Number Theory, wtg/addnoth.notes.dvi. [3] N. Lyall and Á. Magyar, Polynomial configurations in difference sets, to appear J. Number Theory. [4] H. L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, CBMS Regional Conference Series in Mathematics, 84. [5] A. Sárzözy, On difference sets of seuences of integers III, Acta Math. Acad. Sci. Hungar. 3 (978), pp