Type: Article
Publication Date: 2020-04-04
Citations: 18
DOI: https://doi.org/10.1112/plms.12309
We prove a Green–Tao type theorem for multiplicative functions. Asymptotic results on expressions of the form 1.1 are known in many special cases: Green and Tao 14 establish such results for the Möbius function μ. Using the machinery from 13, 14, the author proved asymptotic results for the divisor function d in 26 and, in 25, 27, for the function r counting representations by sums of two squares (and generalisations thereof to representations by binary quadratic forms). Lachand 21 considers the characteristic function of x 1 / u -smooth numbers. The problem of finding upper bounds or even asymptotics for 1.1 in the case where s = 1 is significantly harder and includes the famous open problem of finding the correct asymptotic behaviour of D h ( N ) = ∑ n ⩽ N d ( n ) d ( n + h ) d ( n − h ) as a function of h. When averaging over h ∈ { 1 , … , H } with H = N , this question takes the form 1.1 with s = 2 . Browning 3 showed that the average ∑ h ⩽ H D h ( N ) can be asymptotically evaluated for smaller values of H, more precisely for H ⩾ N 3 / 4 + ε . Using spectral methods, Blomer 2 proves an asymptotic formula for (a smooth version of) ∑ h ⩽ H ∑ n ⩽ N a ( n ) d ( n + h ) d ( n − h ) , where H ⩾ N 1 / 3 + ε and where a is an arbitrary complex-valued arithmetic function. Finally, in work improving on many previous results (see the references in 24), Matomäki, Radziwiłł and Tao 24 recently proved asymptotic formulae for expressions of the form ∑ X < n ⩽ 2 X f ( n ) g ( n + h ) valid for almost all | h | ⩽ H with H ⩽ X 1 − ε and where each of f and g can be a higher divisor function d k , k ⩾ 2 , or the von Mangoldt function1. Our aim here is to leave the pretentious setting and prove a general asymptotic result for 1.1 that applies to unbounded multiplicative functions such as d or r, as well as to functions with small mean values. To illustrate the latter goal, let us mention two examples of functions that the result applies to. Firstly, the result applies, yielding a genuine main term, when we let each h i in 1.1 be a function of the form h i : n ↦ δ ω ( n ) , where δ ∈ ( 0 , 1 ) and ω ( n ) = # { p : p | n } . Such a function h i has a small mean value in the sense that N − 1 ∑ n ⩽ N h i ( n ) ≍ ( log N ) − 1 + δ = o ( 1 ) and the main term in our result will carry the correct logarithmic factor to capture this behaviour. A second example of an admissible function of small mean is h i : n ↦ b ( n ) , where b denotes the characteristic function of sums of two squares. In this case it follows from Landau's work 22 that N − 1 ∑ n ⩽ N h i ( n ) ∼ ( log N ) − 1 / 2 . To handle functions with small mean values, we prove an asymptotic formula with an error term that, instead of being o ( 1 ) , reflects the order of the mean values N − 1 ∑ n ⩽ N | h i ( n ) | for 1 ⩽ i ⩽ r . This allows us to obtain asymptotic results with genuine main term in, amongst others, the above two examples. We proceed by introducing the precise classes of multiplicative functions our main result will apply to. Definition 1.1.Let F denote the class of multiplicative functions h : N → C with the following properties: Together with F, we consider the following slightly larger class F ∗ ⊃ F that contains n i t -twists of elements of F. In other words, condition (iv) only needs to hold for an n i t -twist of f, but not necessarily for f itself. Definition 1.2.Let F ∗ denote the class of multiplicative functions h : N → C that satisfy conditions (i)–(iii) of Definition 1.1, as well as the following variant of condition (iv). The classes F and F ∗ of functions are closely related to the classes F H and F H , n i t studied in 28. Here, we impose the additional assumption (ii) which states that the growth of the function h is bounded like the growth of the divisor function. Out of the four conditions above, the last one is perhaps the least intuitive one and the one that is most difficult to check in an application. Conditions (iv) and (iv) ′ have been studied in detail in [28, § 4], where we prove several sufficient conditions for them. These ought to be significantly easier to check in many applications as they either only involve the values of h at primes or only require to bound the correlation of h with certain Dirichlet characters. To conclude our discussion of the functions classes relevant to this paper let us record some explicit examples of functions satisfying the abstract set of rules defining F. Examples.Elements of the class F include the following functions (cf. [28, §§ 4.2 and 4.3]): The main result of this paper (Theorem 2.4) establishes an asymptotic formula for 1.1 under the assumption that h 1 , … , h r all belong to F ∗ . The proof of this result proceeds via Green and Tao's nilpotent Hardy–Littlewood method (see 13), which is a method consisting of two main parts. One of them requires us to establish that a ‘ W-tricked’ version of any h ∈ F ∗ is orthogonal to arbitrary nilsequences. The other part amounts to showing that this W-tricked version of h has a majorant function which is pseudo-random in the sense of 13 and of the ‘correct’ average order in a sense we will specify later. The first task, namely that of finding a suitable W-trick and obtaining non-trivial estimates for the correlation of the W-tricked version of h and nilsequences, has been established in 28 for all h ∈ F ∗ . The second task, namely the construction of correct-order pseudo-random majorants, will be the main focus of this work. The fact that we are seeking a majorant function for h means that our work is closely related to the sequel of papers studying upper bounds on expressions of the form 2.3 that were referred to at the very start of this introduction. This paper is organised as follows. In Section 2, we give the precise statements of our main result and an easier special case of it. Section 3 describes a reformulation of the main result in terms of short character sums, allowing one in special cases to deduce local-global principles. As an application, we recover a result of Frantzikinakis–Host 9 concerning the case where in 1.1 all the arithmetic functions h 1 , … , h r are ‘pretentious’. Sections 4 and 5 are concerned with the more technical parts of the proof of the main result: Section 4 (conditionally) proves a ‘ W-tricked’ version of the main result, in which each of the multiplicative functions h j from 2.3 is replaced by a function of the form n ↦ h j ( W j n + A j ) for a suitable integer W j and a reduced residue A j ( mod W j ) . To prove this result, we preliminarily assume the existence of families of pseudo-random majorants for the W-tricked functions n ↦ h j ( W j n + A j ) . In Section 5, we then deduce the main theorem from its just established W-tricked version. Sections 6–10 are independent from all preceding sections, apart from the definitions made in this introduction, and contain the main new input of this paper, namely the construction of the required families of pseudo-random majorants for all ( W-tricked versions of the) multiplicative functions from F ∗ . The construction itself takes place in Sections 6–8. Here, the main difficulty lies in establishing suitable majorant functions of the correct average order for bounded multiplicative function, see Section 8. In Section 9 we recall and introduce all relevant concepts around the notion of families of pseudo-random majorants. The task of checking that the constructed majorant functions do indeed give rise to a family of pseudo-random majorants is carried out in Sections 9 and 10. Section 11, finally, contains the proofs of the results from Section 3, and Section 12 discusses the application of our main result to the arithmetic functions h j ( n ) = | λ f j ( n ) | , where λ f j ( n ) denotes the normalised Fourier coefficients of a primitive holomorphic cusp form f j . This section contains the precise statement of our main result, which we present subsequently to that of an easier, but important, special case. We begin by stating the special case of the main result where h 1 , … , h r ∈ F . The restriction to F simplifies the asymptotic formula significantly. This version of the result applies to, amongst others, all the examples mentioned at the end of Section 1.1. Theorem 2.1.Let N > 1 be an integer parameter and let r , s , L ⩾ 2 and B 0 ⩾ 0 be fixed integers. Suppose further that we are given the following data. Remark 2.2. (On removing the ε)The dependence on ε is an artefact of the generality of the result. In cases where the right-hand side of 2.1 can be reformulated as a closed expression that is independent of W ∼ and B 2 , this dependence can be removed. Such a reformulation always exists if for each j ∈ { 1 , … , r } the set of primitive characters χ for which x − 1 | ∑ n ⩽ x h j ( n ) χ ( n ) | is (close to) maximal does not depend on x as soon as x is sufficiently large. (This assumption allows one to replace S h j ( N , W ∼ , A j ) by a short character sum involving a fixed set of characters.) We will describe this set-up in Theorem 3.5. Remark 2.3. (On the parameter T)In applications it is often essential that the one can vary the cut-off parameter slightly while preserving the shape of the main term as well as uniformity in the error term. For this reason we introduced a second cut-off parameter T that is closely related to the value of N which determines W ∼ ( N ) . The full version of our main result extends the class of admissible functions to F ∗ . Compared with the statement of Theorem 2.1, part ( i ) from the assumptions and the asymptotic formula 2.1 itself have to be adjusted. In order to simplify the asymptotic formula, we also slightly change the assumptions in part ( ii ) . This leads to Theorem 2.4. (Main theorem)Let N > 1 be an integer parameter, let r , s , L ⩾ 2 and B 0 ⩾ 0 be integers, let δ ∈ ( 0 , 1 ) be a parameter, fix a value of c ∈ ( 0 , 1 ) and let C > 1 be a constant. Suppose further that we are given the following data. Then there are positive constants B 1 , B 2 = O r , s , B 0 , H , L , α , δ ( 1 ) and a function W ∼ : R > 0 → N , depending on h 1 , … , h r but not on the information from (ii) and (iii), such that This section describes a reformulation of Theorem 2.4 in terms of short character sums, allowing one in special cases to reinterpret the main term in the asymptotic formula as a product over local factors. Corollary 3.1.Let C > 1 and h 1 , … h r be as in Theorem 2.4 and assume that, as N → ∞ , As an example of a function for which 3.1 is not determined by χ 0 but nonetheless by finitely many characters, we may consider the function h ( n ) = 1 4 r ( n ) = 1 ∗ χ − 1 ( n ) , where χ − 1 denotes the non-principal character modulo 4 and where r is the function that counts representations by sums of two squares. In situations where 3.1 is determined by only finitely many characters, our main theorem can be reformulated in terms of short character sums, and Theorem 3.5 is such a reformulation. The fact that the character sum can be truncated relies on the following consequence of the repulsion of characters phenomenon described in 1 and refined in 10, 11. Proposition 3.2. (Set of characters)Suppose that h : N → C is multiplicative and satisfies conditions (i)–(iii) of Definition 1.1. Let t ∈ R and set h ∗ : n ↦ h ( n ) n − i t . Let H ⩾ 1 and α h > 0 be such that parts (i) and (iii) of Definition 1.1 hold, and define a multiplicative function h ′ : N → C by setting h ′ ( p ) = h ∗ ( p ) / H at primes and h ′ ( p k ) = 0 if k ⩾ 2 . Further, let ε = 1 2 min ( 1 , α h / H ) and define the integer k = ⌈ ε − 2 ⌉ . Let C > 0 be a parameter, consider for any x > 1 the set of primitive characters of conductor at most ( log x ) C , and enumerate them as χ 1 , χ 2 , … in such way that the averages | 1 x ∑ n ⩽ x χ i ( n ) h ′ ( n ) | are in non-increasing order as i increases. We let E ( x , C ) : = { χ 1 , χ 2 , … , χ k } denote the set of the first k characters. Define for any given value of N > 1 the set E N = ⋃ 1 ⩽ j ⩽ k ′ E ( N 1 / 2 j , C ) , where k ′ = ⌈ log 2 ( 4 H ) ⌉ . If E N ( q ) denotes the set of characters modulo q induced from elements of E N , then Moreover, we have h ∈ F ∗ , provided for every C > 0 there exists a function φ C : N → R ⩾ 0 with φ C ( x ) → 0 as x → 0 such that Remark 3.3.Note that the statement above simplifies if h ′ is such that the sets E ( x , C ) are independent of x as soon as x is sufficiently large. In this case E N is just given by the fixed set E ( x , C ) = { χ 1 , … , χ k } of the first k characters in the sequence for any C and any sufficiently large x. The finite set of characters picked out by the proposition above may still be larger than strictly necessary in order for 3.3 to hold, and results by Elliott 6, Tenenbaum 32 or Mangerel 23 relating the mean value of a multiplicative function h to that of | h | can be used to further restrict the set E N ( q ) . To illustrate the character of these comparison results, we include the following qualitative lemma, which is a straightforward consequence of Elliott [6, Theorems 2 and 4]. In fact, the first part of this lemma follows already from earlier work of Elliott and Kish [7, Lemma 21]. Lemma 3.4. (Elliott, Elliott–Kish)Suppose h is a multiplicative function that satisfies conditions (i) and (iii) from Definition 1.1 and ∑ p ⩽ H ∑ k ⩾ 2 | h ( p k ) | p − k < ∞ . Then Theorem 3.5.Let N, T and φ 1 , … , φ r be as in Theorem 2.4. For each j ∈ { 1 , … , r } , let h j : N → C be a multiplicative function that satisfies conditions (i)–(iii) of Definition 1.1, let t j ∈ R and suppose that h j ∗ : n ↦ h j ( n ) n − i t j satisfies condition 3.4 from Proposition 3.2. (In particular, h j ∈ F ∗ .) Let W ∼ : N → N be as in Theorem 2.4. For each j ∈ { 1 , … , r } and every sufficiently large N, consider for h = h j the set E j , N : = E N of primitive characters described in Proposition 3.2, and let E j ∗ denote the set of characters modulo W ∼ ( N ) induced from elements of E j , N . Suppose further that for each j and N there is a subset E j , N + ⊂ E j , N and a function ε ( x ) tending to zero as x → ∞ such that as N → ∞ , the following asymptotic formula holds for T ∈ [ N ( log N ) − B 0 , N ] : let B ∈ ( 0 , w ( N ) ] be a cut-off parameter that is sufficiently large in terms of r, H and the bound L on the coefficients of ψ 1 , … , ψ r and let Q denote product of all primes p < B and of the conductors of the characters in E 1 , N + , … , E r , N + ; then, Remark 3.6.Tenenbaum's asymptotic result [32, Theorem 1.3] and the decay estimate given in [32, Corollary 2.1] provide tools for checking the conditions on S h j χ ∗ ( x ) for χ ∈ E j , N + and χ ∈ E j , N ∖ E j , N + in many explicit applications. In those cases where the local factors β p ( χ 1 , … , χ r ) are in fact independent of the characters χ 1 , … , χ r or when, for instance, E j , N is independent of N and # E j , N + = 1 for all j ∈ { 1 , … , r } , then the main term in the previous theorem becomes a product of local factors, allowing one to prove a local-to-global principle. The latter condition holds, for example, when h j is χ j ( n ) n i t j -pretentious, that is, when h j is bounded and 1.2 holds for some character χ j and some t j ∈ R . Asymptotic results for correlations of bounded pretentious multiplicative functions were first proved by Frantzikinakis and Host in [9, Theorem 1.1] and re-proved with explicit main and error terms by Klurman and Mangerel in 20. As a corollary to our main result, we obtain the following version of the pretentious case of these results. Corollary 3.7.Suppose that h 1 , … , h r : N → C are bounded multiplicative functions with the property that for every 1 ⩽ j ⩽ r there exists a character χ j and a real number t j such that ∑ p ( 1 − ℜ ( h j ( p ) χ j ( p ) p i t j ) ) p − 1 < ∞ . Then, as T → ∞ , While the present paper is mainly concerned with the construction of correct-order pseudo-random majorants, no such construction is required in the case of the above corollary, and more generally in the setting of Frantzikinakis and Host's work 9. The reason for this lies in the fact that the trivial majorant given by the all-one function 1 is a pseudo-random majorant of the correct average order for every pretentious multiplicative function. Working with the all-one function 1 as a majorant leads to an error term of the form o ( 1 ) instead of o ( ( log T ) − r ∏ j = 1 r ∏ p ⩽ T ( 1 + p − 1 | h j ( p ) | ) ) in the asymptotic formula 2.3. To be precise, the average order of the majorant appears as a factor in the error term. This is why a majorant of correct average order is required in order to capture the behaviour of functions with small mean values in this asymptotic formula. In this section we prove, assuming the existence of suitable majorant functions, a special case of the main theorem. The main theorem itself will be deduced from this special case in Section 5, while most of the remaining sections of this paper will be concerned with the construction of majorant functions and verification of the required properties: in Sections 6–8 we construct the majorants, in Section 9 we recall and introduce all relevant concepts around the notion of families of pseudo-random majorants and in Sections 9 and 10, we check that the constructed majorant functions give in fact rise to families of pseudo-random majorants. For references purposes, we summarise the results of Sections 6–10 in the statement below, emphasising that all terms are properly introduced in Section 9. Theorem 4.1. (Existence of suitable majorant functions)Let B , D ⩾ 1 be integers, let N > 1 be an integer parameter and suppose W 1 , … , W r ∈ [ 1 , ( log N ) B ] are integers, each divisible by W ( N ) . Let W = lcm ( W 1 , … , W r ) and, for each i ∈ { 1 , … , r } , let A i ∈ { 1 , … , W i } be coprime to W i . Suppose that h 1 , … , h r ∈ F ∗ and define for each i ∈ { 1 , … , r } and for each tuple τ = ( N , W 1 , … , W r , A 1 , … , A r ) the W-tricked function f i ( τ ) : n ↦ h i ( W i n + A i ) together with the weight E f i ( τ ) = E h i ( N ; W i ) . Then there exists a family of D-pseudo-random majorants2 { ν ( τ ) : { 1 , … , ⌊ N / W ⌋ } → R > 0 } τ , for f 1 ( τ ) , … , f r ( τ ) with weights E f 1 ( τ ) , … , E f r ( τ ) . The majorants ν ( τ ) arise as The following special case of Theorem 2.4 works in sub-progressions whose common difference is an integer divisible by the primorial W ( N ) . This procedure removes potential irregularities in the behaviour of the multiplicative functions h j ∈ F ∗ that occur when working in progressions to small moduli. Proposition 4.2.Let B 0 ⩾ 0 and B 2 , C > 0 be constants, let r , s , L ⩾ 2 be integers, let δ ∈ ( 0 , 1 ) and let N , T > 1 be integer parameters satisfying N ( log N ) − B 0 ⩽ T ⩽ N . Let h 1 , … , h r ∈ F ∗ be multiplicative functions and, for each j ∈ { 1 , … , r } , define the function h j ∗ : n ↦ h j ( n ) n − i t j , where t j = t j , N denotes the real number from Definition 1.2 with x = N and the given value of C. Let H > 1 and α > 0 be constants such that conditions (i) and (iii) of Definition 1.1 hold with the given value of H and with α h = α for every h ∈ { h 1 , … , h r } . Then there exist positive constants c ( r , δ ) and B 1 = O B 0 , B 2 , H , α , r , δ ( 1 ) , and, for each N, an integer W ∼ ( N ) ⩽ ( log N ) B 1 , divisible by W ( N ) = ∏ p ⩽ w ( N ) p , such that the following holds, provided C was sufficiently large with respect to H, α, r and c ( r , δ ) . Let φ 1 , … , φ r ∈ Z [ X 1 , … X s ] be linear polynomials as in assumption (ii) of Theorem 2.4 for some fixed value of c ∈ ( 0 , 1 ) , or, if h 1 , … , h r ∈ F , for c = 1 − B 0 log log N . In particular, suppose that the coefficients of the linear forms ψ j = φ j − a j ( N ) are bounded by L in absolute value. Let W 1 , … , W r ∈ [ 1 , ( log N ) B 1 + B 2 ] be integers, each divisible by W ∼ ( N ) and such that W j / W ∼ ( N ) ⩽ ( log N ) B 2 , and let W ′ = lcm ( W 1 , … , W r ) . For each j ∈ { 1 , … , r } , let 0 < A j < W j be an integer coprime to W j . Finally, suppose that K ⊂ [ − 1 , 1 ] s is a convex subset with vol ( K ) > 0 and such that W j φ j ( T W ′ K ) + A j ⊂ [ 1 , T ] for each 1 ⩽ j ⩽ r , all sufficiently large N and all T ∈ [ N ( log N ) − B 0 , N ] . Then, as N → ∞ , we have Remark 4.3.If h 1 , … , h r are such that W ∼ ( N ) and the t j are independent of δ and C, then the term κ ( δ ) in the error term above can be omitted. This is for instance the case when for every j ∈ { 1 , … , r } , both h j ∈ F and when the set of primitive characters χ for which | x − 1 ∑ n ⩽ x h j ( n ) χ ( n ) | is maximal does not depend on x as soon as x is sufficiently large. Observe that Proposition 4.2 is a statement about a family of W-tricked versions of the functions h 1 , … , h r ∈ F ∗ . To prove this result, we will begin by introducing a suitable choice of W ∼ ( N ) , the existence of which is claimed in the statement. Our choice of W ∼ ( N ) will arise from an application of [28, Proposition 5.1]. Once we have set up this application and defined W ∼ ( N ) , we will show that the main result of 28 can in fact be applied to obtain uniform bounds on the correlation with nilsequences that apply uniformly to all W-tricked versions of the h j that arise in the statement of the proposition. This in turn will allow us to apply the machinery from 13, 15 to deduce the proposition. Proof.Let W ∼ ( N ) denote the integer produced by [28, Proposition 5.1] when applied to the collection of functions h 1 , … , h r ∈ F ∗ , with the given values of H and α and with a fixed choice of E > max ( 2 B 2 , B 0 ) . Setting B 1 : = κ + 2 for the value of κ = O E , H , r , α ( 1 ) that arises in the above application of [28, Proposition 5.1], we then have W ∼ ( N ) ⩽ ( log N ) B 1 . The proof of [28, Proposition 5.1] furthermore implies that B 1 = κ + 2 ⩾ E > B 2 . Finally, suppose that the value of C determining the twists h 1 ∗ , … , h r ∗ satisfies C ⩾ 2 E + κ + 4 in accordance with the statement of [28, Proposition 5.1]. Then [28, Proposition 5.1] implies that there exists a function ε : N → R , ε ( N ) = o N → ∞ ( 1 ) such that, writing W = W ∼ ( N ) , The main result of that paper, [28, Theorem 6.1], takes the value of W ∼ ( N ) produced by [28, Proposition 5.1] for any given h ∈ F ∗ and bounds the correlation of n ↦ h ( W ∼ ( N ) n + A ) with nilsequences. The information on W ∼ ( N ) that is used by [28, Theorem 6.1] is exactly the information summarised in the above statement about 4.2 for W = W ∼ ( N ) . Inspecting the range of q in the statement on 4.2, we observe that whenever we set W = w W ∼ ( N ) for some w ⩽ ( log N ) E / 2 , then 4.2 continues to hold for all q ∈ { 1 , ( log x ) E / 2 } and for all intervals I with | I | > x ( log x ) − E / 2 . The fact that we have uniform bounds in 4.2 as W ranges over { w W ∼ ( N ) : w ⩽ ( log N ) E / 2 } , and as h ∗ ranges over { h 1 ∗ , … , h r ∗ } implies that [28, Theorem 6.1] applies uniformly to h ∈ { h 1 , … , h r } and with W ∼ ( N ) being replaced by any integer of the form w W ∼ ( N ) with w ⩽ ( log N ) E / 2 , so long we replace E by E ′ : = E / 2 everywhere in the statement except for in the definition of C. To be precise, this application of [28, Theorem 6.1] yields the following. Let G / Γ be a nilmanifold together with a filtration G • of G of degree d and let g ∈ poly ( Z , G • ) a polynomial sequence. Suppose that G / Γ has a M 0 -rational Mal'cev basis adapted to G • for some M 0 ⩾ 2 and let G / Γ be equipped with the metric defined by this basis. Let F : G / Γ → C be a 1-bounded Lipschitz function. Then [28, Theorem 6.1] implies that, provided E ⩾ 1 is sufficiently large with respect to d, the dimension of G, α and H, we have The bound 4.3 applies in particular to all W j appearing in the statement of Proposition 4.2 since W j / W ∼ ( N ) ⩽ ( log N ) B 2 ⩽ ( log N ) E / 2 by assumption. Thus the function We now seek to apply the (transferred) inverse theorem for uniformity norms from 15 (see [13, Proposition 10.1; 15, Conjecture 1.2 and Theorem 1.3]). Note that this statement only involves linear sequences, that is, their degree is equal to the step of the nilmanifold involved. If M r − 2 , δ denotes the finite set of r − 2 -step nilmanifolds from the statement of the the inverse theorem, let c ( r , δ ) be the maximum dimension of the elements of this set. Then 4.3 applies to all linear sequences associated to manifolds in M r − 2 , δ provided E is sufficiently large with respect to the step r − 2 and c ( r , δ ) . Thus, as soon as N is sufficiently large, it follows from the bound 4.3, from Theorem 4.1 (cf. Theorem 9.2) and from the inverse theorem for uniformity norms from 15 that the function h ∼ j satisfies This allows us to apply the generalised von Neumann theorem [13, Proposition 7.1] (cf. [13, Lemma 4.4]) to deduce that In the case where h 1 , … , h r ∈ F , that is, where t 1 = ⋯ = t r = 0 , the above estimate concludes the proof of the proposition, which could have been simplified in many places. To further simplify the main term of 4.4 in the remaining case where not all t j vanish, our next aim is to show that the summation argument ( W j φ j ( n ) + A j ) i t j can be replaced by ( W j ψ j ( n ) ) i t j . More precisely, using the fact that n i t j varies slowly, we will show that, outside an exceptional set, ( W j φ j ( n ) + A j ) i t j can approximated by ( W j ψ j ( n ) ) i t j . We begin with the exceptional set. Recall that the linear form ψ j = φ j − a j has bounded coefficients and note that for all 0 < T ′ < T we have In this section we deduce Theorem 2.4 from the W-tricked version presented in Proposition 4.2, that is to say, assuming the results about pseudo-random majorants we summarised in Theorem 4.1. Since Proposition 4.2 assumes that the quantities W i , W ∼ ( N ) as well as N / T are all bounded above by a fixed power of log N (an assumption that is essential for the application of the results from 28), the proof of Theorem 2.4 will require us to truncate certain summations. For this purpose we introduce the following exceptional set. Definition 5.1. (Exceptional set)Let D > 0 , N > 1 and let S D ( N ) denote the set all positive integers less than N that are divisible by the square of an integer d > ( log N ) D . Lemma 5.2. (Exceptional set I)Let B 0 , D > 0 and let N , T > 1 be integer parameters with T ⩽ N . Let h 1 , … , h r ∈ F ∗ and let α, H, L and φ 1 , … , φ r ∈ Z [ X 1 , … , X s ] be as in Theorem 2.4. Then Proof.To start with, we have The following simple observation will allow us to discard most cases in which, in the proof of Theorem 2.4, Proposition 4.2 would need to be applied with W j > ( log T ) D for some j ∈ { 1 , … , r } . Lemma 5.3.Let D > 1 and suppose that M ⩽ ( log N ) D is an integer. Then every integer w ∈ ( ( log N ) 3 D , N ] composed entirely out of primes dividing M has a square divisor of size at least ( log N ) 2 D and, hence, belongs to S D ( N ) . Proof.Factorising w as w 1 w 2 2 for a square-free integer w 1 , we have w 1 ⩽ M ⩽ ( log N ) D . Hence, w 2 2 > ( log N ) 2 D . □ Following these preparations we are now ready for the deduction