Arithmetic progressions in Salem-type subsets of the integers
aa r X i v : . [ m a t h . N T ] J u l Arithmetic progressions in Salem-type subsets of the integers
Paul Potgieter
Department of Decision Sciences, University of South AfricaP.O. Box 392, Pretoria 0003, South Africa [email protected]
Abstract
Given a subset of the integers of zero density, we define the weaker notion of the fractionaldensity of such a set. It is shown how this notion corresponds to that of the Hausdorff dimensionof a compact subset of the reals. We then show that a version of a theorem of Laba and Pramanikon 3-term arithmetic progressions in subsets of the unit interval also holds for subsets of theintegers with fractional density which also satisfy certain Fourier decay conditions.Mathematics Subject Classification: 42B05, 11B25, 28A78,26E35
The existence of 3-term arithmetic progressions in certain sets of fractional Hausdorff dimensionwas recently established by Laba and Pramanik [3]. They introduce Salem-type sets in [0 , N and subsets of [0 , R , resembles the usual definition of density very closely, it seemed likely that a weakeridea of density would prove useful in studying arithmetic progressions, especially in the light of [3].Indeed, when subsets of N are mapped to subsets of [0 ,
1] via a mapping similar to that in [4],this “fractional density” is preserved as Hausdorff dimension. Similarly, when a subset of [0 ,
1] ismapped into N , Hausdorff dimension is preserved in the guise of fractional density.The third section discusses a uniformity condition (see for instance [8], p161) necessary for a setof fractional density to contain a 3-term arithmetic progression. In the fourth section, a version of1aba and Pramanik’s result is proved for subsets of N . The proof involves little else but repeateduse of Varnavides’s theorem, as found in [8]. In the final section we construct an example of a set inthe integers, analogous to that found in Section 6 of [3], which satisfies the conditions of Theorem4.1 of this paper.Some background in nonstandard analysis is required for the second section of this paper. Asuccinct but sufficient introduction to all the necessary concepts can be found in [6]. Apart fromDefinition 2.2, the rest of the paper can be read independently of this section. However, in orderto understand the motivation behind the formulation and the direction of future investigations, itwould benefit the reader to at least give it a cursory glance.The author would like to thank Claudius du Plooy for his generous support during the concep-tion of this paper, as well as for many enlightening conversations throughout the years. N and [0 , We use the notation of [6] throughout. Let A = ( a n ) n ∈ N denote a sequence of natural numbers(which we assume to be strictly increasing). Note that in the paper [4] we are not restricted tosequences in N , but it will suffice for our purposes. The essential idea behind the correspondence isto use a hyperfinite number to divide every member of the nonstandard extension ∗ A of the sequence A (throughout this section we denote nonstandard extensions of sets similarly). The standard partof a nonstandard number or set x shall be denoted by st( x ). General results in [4] hold for divisionby any hyperfinite number z ∈ ∗ N \ N . We shall however only consider division of each elementof ∗ A by the number h a n i U , that is, the unique hyperfinite number determined by the sequence( a n ) under the equivalence relation of a certain (fixed) free ultrafilter U (the choice of ultrafilter isimmaterial to the results). We formalise this previous by defining: Definition 2.1.
Suppose A = ( a n ) n ≥ is an increasing sequence of natural numbers. Then wedenote by st z ( A ) the set { st ( a/z ) : a ∈ ∗ A } where z = h a n i U . (The definition can of course be extended from N to Z .) It is clear that st z ( A ) ⊆ [0 , st z ( A ) coincides with the “fractional density” of A . Throughoutthe paper we will use [ A, B ] to denote the interval in Z given by the set { A, A + 1 , . . . , B } . Theintervals [ A, B ) and (
A, B ] are defined analogously.
Definition 2.2.
We say that a set A ⊆ N has fractional upper density α if lim sup N →∞ | A ∩ [1 , N ] | N β is ∞ for any β < α and for any β > α . (The lower fractional density can be similarly defined by replacing lim sup in the above withlim inf. If upper and lower fractional densities are equal, we can just speak of the fractional density.)We can summarise this by saying that ¯ d f ( A ) = α . We will sometimes need to consider thefractional density relative to a finite but arbitrarily large number; that is, we will say that A ⊆ [0 , N )2as upper fractional density β relative to N if | A | /N β = c >
0, for arbitrarily large N . In sections3 and 4, this is mostly how the concept of upper fractional density will be utilised. Note that thelimit in the above definition is the same as the limit of n/a βn , which is the form we will use inproving Proposition 2.2.Of course, one has to verify that such a concept yields information that the usual definition ofdensity does not, in the same way that Hausdorff dimension (denoted by dim H ) yields informationthat Lebesgue measure does not. Firstly, it is easily verified that any subset of N of positive densityhas fractional density 1. One can also verify that there exist sets which do not have positive densitybut do have positive fractional density. For example, one can create a version of the triadic Cantorset on N as follows:1. Let C be the interval (0 , ] (in N ). We recognise only the right-hand endpoint of the interval,leaving C = { } .2. Let C consist of the interval [0 , ]. Remove the middle third (1 , C = { }
3. Similarly with the interval (0 , ], we remove the middle third intervals (1 , , , ,
6] and (7 , C = { , , , } , and so on.This construction can be formalised thus: C = { } (2.1) C i +1 = C i ∪ { i +1 + 1 − c : c ∈ C i } , i ∈ N (2.2) C = ∞ [ i =0 C i . (2.3)It is trivial to show that this set has fractional density log 2 / log 3 simply by counting elementsat every stage, even though it does not have positive density.Instead of utilising the standard definition of Hausdorff dimension, we use the following non-standard version [6]. Note that for some infinitesimal △ t = 1 /N , N ∈ ∗ N \ N , we call the set { , △ t, △ t, . . . , − △ t } the hyperfinite time line based on △ t . The function | · | denotes thetransferred cardinality function. Theorem 2.1.
Consider a hyperfinite time line T based on the infinitesimal N − , for a given N ∈ ∗ N \ N . Suppose that an internal subset A ′ of the time line is such that ◦ ( A ′ ) = A and for some α > ◦ (cid:18) | A ′ | N β (cid:19) > for β < α and (2.4) ◦ (cid:18) | A ′ | N β (cid:19) = 0 for β > α. (2.5) Then α = dim A . One might be concerned that the nonstandard formulation of Hausdorff dimension might tooclosely resemble Minkowksi dimension. However, as is shown in [6], this formulation implies theexistence of a positive measure on a set of positive Hausdorff dimension, which is not necessarily aproperty of sets of positive Minkowski dimension.3 simple argument using the transfer principle now shows that fractional density of the set A is exactly the same as the Hausdorff dimension of the set st z ( A ). Proposition 2.2.
Suppose that a sequence ( a n ) = A ⊆ N has fractional density α . If z = h a n i U ,then st z ( A ) has Hausdorff dimension α . Proof. If β < α , the sequence n/a βn will diverge as n → ∞ . Hence we can assume that for all i after a certain stage, i/a βi >
1. If we now let a J denote the element of the nonstandard extension ofthe sequence determined by the sequence itself (modulo the free ultrafilter), the property J/a βJ > { a/a J : a ∈ ∗ A } , we see that it is asubset of the hyperfinite time line based on a J , since each element of ∗ A is still a member of ∗ N (by transfer). Furthermore, Theorem 2.1 now implies that dim H ( st a J ( A )) > β . Similarly, for any β > α , we obtain that dim H ( st a J ( A )) < β , concluding the proof.The converse of the previous proposition can also be easily shown by reversing the argument,i.e. that given a subset of [0 ,
1] of Hausdorff dimension α , we can multiply by a hyperfinite naturalnumber (which is not unique) to obtain a set with fractional density α . A more interesting questionconcerns the relationship between the Fourier-dimensional properties of compact sets in R and theproperties of discrete Fourier coefficients of characteristic functions of analogous subsets of Z . Itis this relationship we are attempting to explore by interpreting the results in [3] in the context ofthe whole numbers. The essence of the proof of Roth’s theorem, as presented in e.g. [5], is to show that the Fouriertransform of the characteristic function of a set of positive density either satisfies certain decayconditions, or the set has increased density in some arithmetic progression in Z . Iterating thisargument on the assumption that the set contains no 3-term arithmetic progressions, a density ofgreater than 1 is eventually obtained on some arithmetic progression, a contradiction.If the set does not have positive density, we have to impose decay conditions on the Fouriercoefficients. We first determine the uniform rate of decay necessary to guarantee such progressionswhen a set has fractional density α <
1. We will say that a subset A of a finite additive group Z is γ -uniform if the Fourier coefficients of the characteristic function satisfy | c χ A ( k ) | ≤ γ for all k ∈ Z , k = 0. If this γ is small, the set is said to be linearly uniform. In the case of a set ofpositive density, it is possible to find linear uniformity conditions which guarantee the existence ofprogressions. Our version of this will be to find some β such that if the Fourier coefficients are allsmaller than cN β for some c , we will be guaranteed a 3-term arithmetic progression.Consider A ⊂ Z such that for some 0 < α < | A ∩ [0 , N ) | ≥ δN α for arbitrarily large N . (This implies that the upper fractional density of A is ≥ α .) We will assume, without loss,that | A ∩ [0 , N ) | = δN α for each N under consideration. As a first approximation to the 3-termarithmetic progressions contained A ∩ [0 , N ), we count the number of progressions modulo N , i.e.the number of x, y, z ∈ A such that x + y ≡ z mod N. (In this we follow Lyall’s exposition of Roth’s theorem [5], and use similar notation.) The Fouriercoefficients of a function defined on the integers modulo N (denoted by Z N ) are defined as usual4y ˆ f ( k ) = 1 N N − X n =0 f ( x ) e − πiknN The number of triples satisfying the congruence, if χ A denotes the characteristic function of A , isgiven by N = N N − X n =0 c χ A ( n ) c χ A ( n ) c χ A ( − n )However, a triple satisfying the congruence does not necessarily form a true arithmetic progressionin Z , since some of the terms might “wrap around” the cyclic group. If we require instead that x, z ∈ M A = A ∩ [ N/ , N/ Z N -progression does indeed form a Z -progression. In thiscase, we estimate the true triples N by writing N ≥ N N − X n =0 d χ M A ( n ) c χ A ( n ) d χ M A ( − n ) = δN α − | M A | + N N − X n =1 d χ M A ( n ) c χ A ( n ) d χ M A ( − n ) . We require that | M A | ≥ δ N α and | c χ A ( k ) | ≤ δ N β /
32 for k = 0. Using the Cauchy-Schwartzinequality, this gives N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N − X n =1 d χ M A ( n ) c χ A ( n ) d χ M A ( − n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ N max k =0 | c χ A ( k ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N − X n =1 d χ M A ( n ) d χ M A ( − n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ N max k =0 | c χ A ( k ) | N − X n =1 | d χ M A ( n ) | ! X n | d χ M A ( − n ) | ! ≤ N max k =0 | c χ A ( k ) | N − X n =0 | d χ M A ( n ) | = N max k =0 | c χ A ( k ) | · N N − X x =0 χ M A ( x ) ≤ δ N β · N · | M A |≤ δ N β N α +1 If we now require that β < α −
2, say β = 2 α − − ε , we find that N ≥ δN α − | M A | − δ N α − − ε ≥ δ N α −
32 (2 − N − ε ) > δ N α − α > /
3. 5e still have not taken into account the number of trivial progressions x = y = z , of whichthere are | A | = δN α . If we subtract this from the estimate obtained above and require that α > / N > /δ , we are certain to have a non-trivial 3-progression.Of course, we might not always be as fortunate as to have such small Fourier coefficients. Inthe next section we show that weaker non-uniform conditions would still suffice, provided that thedecay is sufficiently structured. In this section we prove the following:
Theorem 4.1.
Let A ⊆ Z . Suppose A satisfies the following conditions:(i) A has upper fractional density α , where α > / .(ii) The Fourier coefficients of the characteristic functions χ A N of A N = A ∩ [0 , N − satisfy | d χ A N ( k ) | ≤ C ( | k | N ) − β/ for large N , for some / < β ≤ satisfying β > − α .Then A contains an arithmetic progression of length 3. As long as the interval [0 , N −
1] is fixed, as it is throughout most of the proof, we will usesimply A instead of A N .To prove Proposition 4.1, we use a modified version of the density arguments using Varnavides’stheorem, to be found in e.g. [8]. Throughout, we use Z to denote a finite additive group of oddorder N . The expectation of a function on Z is defined as E Z ( f ) = E x ∈ Z ( f ) = 1 | Z | X x ∈ Z f ( x )The L p ( Z )-norm of a function f : Z → C is given by k f k L p ( Z ) = N N − X n =0 | f ( n ) | p ! p . We also define the linear bias of a function f : Z → C by k f k u ( Z ) = sup ξ ∈ Z | ˆ f ( ξ ) | . In the proof we will repeatedly use the following definition:
Definition 4.1. Λ ( f, g, h ) = E x,r ∈ Z f ( x ) g ( x + r ) h ( x + 2 r )Note that | Z | Λ ( χ A , χ A , χ A ) is an indication of the number of 3-term arithmetic progressionsto be found in a set A ⊆ Z , although some might be counted more than once. To remove trivialprogressions, one has to subtract | A | . It follows that if | Z | Λ ( χ A , χ A , χ A ) − | A | is suitably large, A will contain at least one 3-progression as a subset of the group Z .The following can be found in [8], p.374. 6 roposition 4.2. For functions f , g and h from Z to C , Λ ( f, g, h ) = N − X n =0 b f ( n ) b g ( − n ) b h ( n ) . (4.6)We also have the following property of Λ [8]:Λ ( f, g, h ) ≤ k f k u ( Z ) k g k L ( Z ) k h k L ( Z ) (4.7) Proof of Theorem 4.1.
We can assume that not all of the Fourier coefficients are smallerthan or equal to δ / N β , since that would immediately imply a 3-term arithmetic progression, bythe result in Section 3.From now on, we denote χ A by µ , for brevity and also to consolidate the analogy with [3]. Wherethey consider a compact subset of [0 ,
1] of certain Hausdorff dimension together with a sufficientdecay of the measure guaranteed to exist on the set, we consider a set of fractional density withsufficient decay of the discrete Fourier transform of the characteristic function.We decompose µ into a sum µ + µ . Using this, we estimate the expression Λ( µ, µ, µ ). If thisis large enough, it will guarantee the existence of a 3-term arithmetic progression.We let F K denote a version of the Fej´er kernel on [0 , N − F K ( x ) = K X k =0 (cid:18) − kK + 1 (cid:19) e πikxN . Define µ as the convolution of µ and F K : µ ( x ) = ( F K ∗ µ )( x ) = N − X y =0 K X n =0 (cid:18) − nK + 1 (cid:19) e πin ( x − y ) N µ ( y ) . By rewriting the convolution product, we can find the Fourier series of µ : µ ( x ) = K X k =0 N − X n =0 (cid:18) − kK + 1 (cid:19) e πikxN e − πiknN χ A ( n )= K X k =0 (cid:18) − kK + 1 (cid:19) e πikx c χ A ( k ) . Thus, if n < K + 1, c µ ( n ) = (cid:18) − nK + 1 (cid:19) c χ A ( n ) . Otherwise, c µ ( n ) = 0. Also, since c µ ( n ) = c χ A ( n ) − c µ ( n ), c µ ( n ) = min (cid:18) , nK + 1 (cid:19) c χ A ( n ) . To calculate Λ ( µ, µ, µ ), we split the expression Λ ( µ + µ , µ + µ , µ + µ ) into eight termsof the form Λ ( µ i , µ j , µ k ), i, j, k ∈ { , } . The idea is then to show that the term Λ ( µ , µ , µ )dominates the others, and will be large enough to guarantee an arithmetic progression.7e can now use the following inequality, which follows from (4.2): | Λ ( f, g, h ) | ≤ X ≤ n 2, so that, for 1 ≤ n ≤ K , | − n | − β ≤ (2 n ) − β . Furthermore, this implies that min { , − ( N − n ) / ( K + 1) } = 1 for 1 ≤ n ≤ K . We willlater see that the lower bound we place on K does not violate these conditions. This assumptionallows us to replace | − k | by 2 k in the sequel.First considering the term Λ ( µ , µ , µ ), we know from inequality 4.8, the fact that c µ ( n ) = 0for n ≥ K + 1 and c µ (0) = 0 that | Λ ( µ , µ , µ ) | ≤ X 8e approximate the expressionΛ ( µ , µ , µ ) = Λ ( µ + µ , µ + µ , µ + µ )by showing that one term is large compared to the seven others.It is clear that Λ ( µ , µ , µ ) = δ N α − . Furthermore, c µ (0) = 0 and c µ ( k ) = c µ ( k ) for k > | Λ ( µ , µ , µ ) | ≤ k µ k u ( Z ) k µ k L ( Z ) k µ k L ( Z ) = δN α − (cid:20) max n (cid:26)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) − nK + 1 (cid:19) c χ A ( n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:27)(cid:21) k µ k L ( Z ) = O ( δN α − β − k µ k L ( Z ) ) . Assuming that K = O ( N ), we can use Parseval and an integral to approximate the L ( Z )-norm of µ : k µ k L ( Z ) ≤ N − X n =0 | c µ ( n ) | = O K X n =1 (cid:18) − nK + 1 (cid:19) k − β N − β ! = N − β O "Z K (cid:18) − x − K + 1 (cid:19) ( x − − β dx + (cid:18) − K + 1 (cid:19) = O ( N − β + ) . Therefore, k µ k L ( Z ) = O ( N − β + ). It follows that the term | Λ ( µ , µ , µ ) | is O ( N − + α )(remembering that β > − α , and the same clearly holds for | Λ ( µ , µ , µ ) | . Similar calculationsshow that similar upper bounds hold for every term involving µ . Since α > / 2, this is smallcompared to N α − .All of the approximations now imply thatΛ ( µ, µ, µ ) = Ω( N α − ) . The number of arithmetic progressions in Z is counted by the expression N Λ ( χ A , χ A , χ A ) − | A | where the second term is employed to ensure we disregard progressions with difference 0. It isimportant to observe here that the progressions counted is the number of proper progressions (i.e.with non-zero difference) in the cyclic group Z , which may not be equivalent to the number ofprogressions in the interval [0 , N − ⊂ Z (which will be referred to as genuine progressions).The question is now how to eliminate the progressions which “wrap around” the cyclic group Z .In Roth-type theorems, this is often done through density-increment arguments, for instance inchapter 10 of [8]. In our case, we instead consider the set A as a subset of the interval [0 , N ),9hich we can again consider as a cyclic group, which we will call Z ′ . (This is an embedding of A N into [0 , N ), not a restriction of the original set to a larger interval.) Any proper progression in A ,seen as a subset of Z ′ , would now have to be a genuine progression, since there are no elements of A in the interval [ N, N ). Assuming that there are no progressions except trivial ones, this meansthat the total number is simply the cardinality of A .We still denote the characteristic function of A as a subset of Z by χ A , whereas the characteristicfunction of A as a subset of Z ′ is denoted by χ A ′ . The effect on the Fourier coefficients of χ A isto “smear” them in such a way that their contribution to the sum-of-squares in the Parsevalinequality is taken up by several Fourier coefficients of χ A ′ . By simply using the definition of theFourier coefficients, it is easily shown that | d χ A ′ (3 k ) | + | d χ A ′ (3 k − | + | d χ A ′ (3 k − | ≤ | c χ A ( k ) | . This now has the implication that A ′ satisfies condition (ii) of Theorem 4.1, with some slightlymodified constants. It is also obvious that A ′ has the same fractional density α as A . Thus, theproof implies that the number of three-term arithmetic progressions in A ′ is greater than cN α − − | A ′ | for some constant c . Since all progressions counted by this expression are genuine, we have estab-lished the existence of the required progressions in A . In this section we present a version in the whole numbers of the Salem-type set constructed in [3].Consider the set { , , , . . . , N j − } for N and j large, and some t , 1 ≤ t ≤ N . Our aim is toconstruct a set which has fractional density α = log t/ log N (relative to the finite set N j ) and forwhich the Fourier coefficients of the characteristic function satisfy condition (ii) of Proposition 4.1,with β > − α . At each of the j stages of the construction, we randomly pick a number of pointsfrom the total in a ratio t/N , in such a way that the Fourier coefficients of successive sets satisfycertain inequalities.Let A = { , , . . . , N j − } . Divide A into N equal intervals (in the whole numbers, as usual)of length N j − . Let the left-hand endpoints of these intervals be denoted by B ∗ = { , N j − , N j − . . . , ( N − N j − } . From this set we choose t elements with equal probability 1 /t , and call this B . We form A fromthis by setting A = [ b ∈ B { b, b + 1 , . . . , b + N j − − } . We now divide each interval of A into N equal pieces of length N j − and form the set B ∗ = [ b ∈ B { b, b + N j − , . . . , b + ( N − N j − } from the endpoints of the intervals newly divided. For each of the t components in the unionconstituting B ∗ , we now have N elements, and from each choose t uniformly and call the resulting10random) set B . The choice of t elements associated to an element b of B ∗ we call B x ( b ) , whilst theportion of B ∗ of length N j − starting at b is denoted by B ∗ ,b . Iterating this construction, we obtainfrom a set A m consisting of t m intervals of length N j − m , a subdivision characterised by B ∗ m +1 anda choice of t m +1 subintervals characterised by B m +1 , which we then use to obtain A m +1 .Some quick calculation will show that this set has fractional density log t/ log N relative toeach interval [0 , N j ). In order to determine the rate of decay of the discrete Fourier transform, weborrow the technique utilised in [3], pp. 20–26, adapted to the whole numbers. Fundamental tothe calculation is a version of Bernstein’s inequality by Ben Green [1]. Lemma 5.1. Let X , . . . , X n be independent random variables with | X j | ≤ , E X i = 0 and E | X j | = σ j . Let P σ j ≤ σ , and assume that σ ≥ nλ . Then P (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X X j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ nλ ! ≤ e − n λ / σ . Given a set B ⊂ [0 , S B ( k ) = X b ∈ B e − πikb . If we are instead considering a set B ⊂ Z with B ⊂ [0 , N j ), we abuse the notation by also using S B ( k ) to denote the sum X b ∈ B e − πikbNj . In this way, we can either regard S B as an exponential sum, or as the Fourier transform of thecharacteristic function multiplied by a factor N j .The previous lemma can be used to prove the following, which is a restatement of Lemma 6.2in [3]: Lemma 5.2. Let B ∗ = { , MN , MN , . . . , N − MN } and let ≤ t ≤ N . Let η t = 32 log 8 N M Then there exists a set B ( x ) ⊂ B ∗ with | B | = t such that (cid:12)(cid:12)(cid:12)(cid:12) S B ( x ) ( k ) t − S B ∗ ( k ) N (cid:12)(cid:12)(cid:12)(cid:12) ≤ η for all k ∈ [0 , M N ) , x ∈ [0 , N − , where B ( x ) = (cid:26) ( x + y ) modNM N : y ∈ B (cid:27) . In the proof of this from Lemma 5.1, it is shown that the condition is satisfied with probabilitygreater than half, indicating that at least half of all possible choices of B ( x ) will have the property.One more tool will be necessary before we start the proof – an approximation of the Fouriercoefficients by an integral. Specifically, by considering the integral of a smooth function f : R → C from a to b as being approximated by a left Riemann sum with step-size ∆ = ( b − a ) /M , we get11 (cid:12)(cid:12)(cid:12)(cid:12)Z ba f ( x ) dx − ∆ M − X n =0 f ( a + n ∆) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c ( b − a ) M sup x ∈ [ a,b ] | f ′′ ( x ) | , where the constant c is independent of M , a and b .We can now use a proof similar to that in [3], with some adjustment for the error term.Define ψ m ( k ) = N m t m d χ A m ( k ) = N m t m N j X a ∈ A m e − πikaNj ! . Although ψ m is not quite the same as the Fourier transform, it will yield enough information todetermine an upper bound.Let B m be in relation to A m as in the construction above. Then ψ m ( k ) = N m t m X b ∈ B m N j (cid:18) e − πikbNj + e − πik ( b +1) Nj + · · · + e − πik ( b + Nm − j − Nj (cid:19) (5.9)Note that if the left-hand endpoint of a subinterval of length N j − m − is determined, the wholeinterval is determined. If we consider a choice of t numbers from a collection of N numbers todetermine the start of the interval, the exact same choice can be considered to be applied N j − m − times, from a sample space consisting of translates of the N starting points of the intervals. Inthe Fourier transform of the characteristic function of the interval, these terms then contribute thesame as the starting point, except for a phase shift for each element. If we now wish to computethe difference | ψ m +1 − ψ m | , the above expression for ψ m shows that we can consider the difference (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N m +1 t m X b ∈ B m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S B ∗ m,b ( k ) N − S B x ( b ) ( k ) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N j N j − m − − X n =0 e − πiknNj (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (5.10)In the above, we stay close to the notation of [3] in denoting the exponential sum over the set B ∗ m,b = { b, b + N j − m − , b + 2 N j − m − , . . . , b + ( N − N j − m − } by S B ∗ m,b and the sum over thecorresponding t -choice by S B x ( b ) . We now approximate the final sum by an integral:1 N j N j − m − − X n =0 e − πiknNj = Z N − ( m +1) e − πikx dx + O k N − m +1) N j ! , where the error term is that of a Riemann sum-approximation of the integral using a step-size N − j .The error term can easily be shown to be less than the integral in absolute value, especiallykeeping in mind that we can choose N arbitrarily large. Hence we dispose of it in the absolutevalue, keeping in mind that it might necessitate the use of a constant c < 2, which is not dependenton m . Computing the integral, we find | ψ m +1 ( k ) − ψ m ( k ) | ≤ c (1 − e − πik/N m +1 ) t m (2 πik/N m +1 ) X b ∈ B m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S B ∗ m,b ( k ) N − S B x ( b ) ( k ) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (5.11)It is now obvious that the above equation is very nearly of the same form as (52) in Lemma 6.4of [3]. We can therefore apply the result of the lemma to obtain | ψ m +1 ( k ) − ψ m ( k ) | ≤ 32 min (cid:18) , N m +1 | k | (cid:19) t − m +12 log(8 N m +1 ) . (5.12)12e now show that the condition 4.1 (ii) is satisfied for any β > α such that β > − α . Since ψ ( k ) = 0 for all k ∈ { , , . . . , N j − } , we can find an upper bound on ψ j ( k ) by bounding the sumof all such differences. By noting that t = N α , we can write the summand as follows (ignoring theconstant factor, which has no bearing from here on):min (cid:18) , N m k (cid:19) t − m log(8 N m ) = min (cid:18) , N m k (cid:19) N − αm (log 8 + m log N ) (5.13)= min (cid:18) , N m k (cid:19) N − βm N − ( α − β ) m (log 8 + m log N ) (5.14)Using the fact that N − ( α − β ) m/ j log N ≤ α − β ) − (which can be established using elementarycalculus [3]), the sum is bounded by j X m =1 min (cid:18) , N m k (cid:19) N − βm (cid:16) N − ( α − β ) m log 8 + 2( α − β ) − (cid:17) ≤ j X m =1 min (cid:18) , N m k (cid:19) N − βm (cid:0) log 8 + 2( α − β ) − (cid:1) (5.15)We consider two different regions: one where 1 ≤ m ≤ log k/ log N and one where m > log k/ log N . In the first case, S = k − t j (log 8 + 2( α − β ) − ) X ≤ m ≤ log k log N N m (1 − β ) (5.16)The sum on the right is easily bounded, thus S ≤ k − (log 8 + 2( α − β ) − ) k − β ≤ C k − β/ (5.17)for some C independent of N , j .Approximating the second part of the sum is similar, and we obtain S = (log 8 + 2( α − β ) − ) X log k log N 1] and sets in Z , which preserves Hausdorff- and Fourier-dimensionalproperties. An examination of the precision of the correspondence will appear in the sequel to thispaper. 13 eferences [1] B. Green, Arithmetic progressions in sumsets , Geom. Funct. Anal. (2002), 584-597[2] B. Green, T. Tao, The primes contain arbitrarily long arithmetic progressions , Ann. Math. (2008), 481-547[3] I. Laba, M. Pramanik, Arithmetic progressions in sets of fractional dimension , Geom. Funct.Anal. (2009), 2, 429-456[4] S.C. Leth, Applications of nonstandard models and Lebesgue measure to sequences of naturalnumbers , Trans. Amer. Math. Soc. (1988), 2, 457-468[5] N. Lyall, Roth’s theorem - The Fourier analytic approach ∼ lyall/REU/Roth.pdf[6] P. Potgieter, Nonstandard analysis, fractal properties and Brownian motion , Fractals (2009), 117-129[7] K. Roth, On certain sets of integers , J. London Math. Soc.28