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M\"{o}bius disjointness for a class of exponential functions

A vast class of exponential functions are shown to be deterministic. This class includes functions whose exponents are polynomial-like or"piece-wise"close to polynomials after differentiation. Many of these functions are proved to be disjoint from the M\"obius function.


Introduction
Let µ(n) be the Möbius function, that is, µ(n) is 0 when n is not square free (i.e., divisible by a nontrivial square), and is (−1) r when n is a product of r distinct primes. Many problems in number theory can be reformulated in terms of properties of the Möbius function. For example, the Prime Number Theorem is known to be equivalent to n≤N µ(n) = o(N). The Riemann hypothesis holds if and only if n≤N µ(n) = o(N 1 2 +ǫ ) for every ǫ > 0. For a truly random sequence a n of −1 and 1, the normalized average ( N n=1 a n )/N 1 2 obeys Gaussian law in distribution as N tends to infinity, which implies that n≤N a n = o(N 1 2 +ǫ ). Under the Riemann hypothesis, the Möbius function shares this property as an indication that certain randomness may exist in the values of the Möbius function. It is widely believed that this randomness predicts significant cancellations in the summation of µ(n)ξ(n) for any "reasonable" sequence ξ(n). This rather vague principle is known as an instance of the "Möbius randomness principle" (see e.g., [20,Section 13.1]). In [36], Sarnak made this principle precise by identifying the notion of "reasonable" and proposed the following conjecture. Here, we recall the definition of deterministic sequences. Functions from N to C are called arithmetic functions or sequences. An arithmetic function f (n) is said to be disjoint from another one g(n) if N n=1 f (n)g(n) = o(N). Let (X , T ) be a topological dynamical system, that is X is a compact Hausdorff space and T : X → X a continuous map. We say n → F (T n x 0 ) an arithmetic function realized in the topological dynamical system (X , T ), and (X , T ) a realization of this arithmetic function. Functions realized in topological dynamical systems of zero topological entropy are called deterministic.
In recent years, a lot of progress has been made on SMDC. See [2,3,4,7,8,9,13,14,16,17,18,22,23,24,25,26,33,35,39,42,43,44], to list a few. In the following, we shall discuss only the results that are more related to this paper. The goal of this article is to show that a class of exponential functions is deterministic and to verify the conjecture of Sarnak for them.

Notation.
We use e(f (n)) to denote the exponential function exp(2πıf (n)) when f is a real-valued arithmetic function, where ı is the imaginary unit. When we say the exponential function of f (n), we mean e(f (n)). Sometimes we call e(f (n)) an f (n) phase.
We use 1 S to denote the indicator of a predicate S, that is 1 S = 1 when S is true and 1 S = 0 when S is false. We also denote 1 A (n) = 1 n∈A for any subset A of N. For any finite set C, |C| denotes the cardinality of C.
We use {x} and ⌊x⌋ to denote the fractional part and the integer part of a real number x, respectively. We use x R/Z to denote the distance between x and the set Z, i.e., x R/Z = min({x}, 1 − {x}). For ease of notation, we drop the subscript and write simply x .
The difference operator △ is defined on the set of all arithmetic functions, mapping f (·) to f (· + 1) − f (·). The k-th difference operator △ k is defined by the composition of △ with k times.
For two arithmetic functions f (n) and g(n), f = o(g) means lim n→∞ f (n)/g(n) = 0; assume that g(n) ≥ 0 for any n ∈ N, f ≪ g means that there is an absolute constant c such that |f | ≤ cg; f = g + O(h) means f − g ≪ h.

1.2.
A class of deterministic sequences. It was investigated in [10] and [41], the set of deterministic sequences is closed under many operations, for example addition, multiplication, inversion, conjugation and translation. This set is also closed under the uniform limit. Moreover, any continuous function of a deterministic sequence is also deterministic.
Motivation. From the above, it is easy to see that the set of deterministic sequences is closed under the difference operator △. As we know, △ has an inverse operator σ (up to an initial value f (0)), mapping f to σ(f ) : n → j<n f (j). Then it is natural to ask whether σ(f ) is also deterministic when f is deterministic. We may assume f is real in this question because a complex arithmetic function is deterministic if and only if both its real part and imaginary part are deterministic. However, σ(f ) could be unbounded (so the corresponding dynamical system is no longer compact) even if f is bounded. It is therefore better to consider what kind of properties f has can ensure that e(σ(f )) is deterministic. For this question, we give the following answer, which states that e(σ(f )) is deterministic when the k-th difference △ k f (n) is "piece-wise" close to polynomials. Theorem 1.2. Let w be a positive integer. Suppose that p 1 (y), . . . , p w (y) are polynomials in R[y], and N = S 1 ∪ S 2 ∪ · · · ∪ S w is a partition of N with each 1 Sv (n) deterministic. Let g(n) = w v=1 1 Sv (n)p v (n). (1.1) Then for any real-valued arithmetic function f (n) satisfying for some k ∈ N, e(f (n)) and e(σ(f )(n)) are deterministic.
Now, we explain briefly the main idea to prove the above result. The major tool we use is anqie entropy (of arithmetic functions), which was introduced by Ge in [10]. We refer readers to Section 2.1 for knowledge on anqie entropy. To prove Theorem 1.2, we first construct a sequence of arithmetic functions {f N (n)} ∞ N =0 with finite ranges that uniformly converges to f (n). By the lower semi-continuity of anqie entropy (see Proposition 2.2), it suffices to show that the anqie entropy of f N is zero for N large enough. Note that the anqie entropy of f N (with finite range) is given through the cardinality of different J-blocks occurring in it (see formula (2.1)). So we focus on estimating this cardinality. Our method is to built a one-to-one map from the set of J-blocks occurring in the sequence f N (n) to the set of pieces of R k cut by hyperplanes (for some k depending on J). So the cardinality of the first set is bounded by the cardinality of the latter one. And we prove that the second cardinality has polynomial growth (see Lemma 2.6). We refer readers to Section 3 for more details.
We also consider whether the exponential function of any concatenation of polynomials is deterministic. Before stating the next result, we first introduce the definition of concatenation of arithmetic functions. Definition 1.3. Let 0 = N 0 < N 1 < · · · be a sequence of natural numbers with lim i→∞ (N i+1 − N i ) = ∞, and {f i (n)} ∞ i=0 be a sequence of arithmetic functions. We say that f (n) is the For the exponential functions of concatenations, we obtain the following result. Theorem 1.4. Let g(n) be defined as in Theorem 1.2. Suppose that {f i (n)} ∞ i=0 is a sequence of real-valued arithmetic functions such that holds for some k ∈ N. Then for any concatenation , e(f (n)) is deterministic. As an application of Theorem 1.4, we prove that Sarnak's conjecture implies the averages of the products of Möbius and certain exponential functions are small in almost all short intervals (Theorem 3.2).

Disjointness of
Möbius from e(f (n)) with the k-th difference of f (n) tending to zero. The disjointness of Möbius from exponential functions is important and has been extensively studied in number theory. For example, the disjointness of µ(n) from e(nα), for any α ∈ R, is closely related to the estimate of exponential sums in prime variables, from which one can deduce Vinogradov's three primes theorem. It is known that µ(n) is disjoint from e(p(n)) for p(n) a polynomial (or a sub-polynomial) (see [6], [15,Chapter 6,Theorem 10] and [40,Chapter 3,Theorem 4.6]). Moreover, the upper bound of N n=1 µ(n)e(p(n)) has received much attention (see e.g., [1], [21] and [46]) partially due to its close connection with the distribution of the zeros of Dirichlet L-functions.
As we have seen from Theorem 1.2 that e(f (n)) is deterministic when f (n) satisfies that there is a k ∈ N such that the k-th derivative (or the k-th difference for the discrete case, see condition (1.4) below) tends to zero. Note that SMDC implies that any deterministic sequence is disjoint from the Möbius function. Motivated by this, we are interested in the following problem.
Problem 1. Let f (n) be a real-valued arithmetic function such that for some natural number k. Is µ(n) disjoint from e(f (n))?
Restrictions on the k-th derivative or k-th difference of f (n) often appear in nontrivial estimate of the exponential sums N n=1 e(f (n)). For example, Van der Corput's method and the method of exponential pairs (see e.g., [11]). The latter one usually requires △f (n) to be approximately cn −s for some c, s > 0. This condition is included in (1.4). In this paper, we focus on the sum N n=1 µ(n)e(f (n)) under the restriction (1.4). In the following we investigate Problem 1 without assuming SMDC. For the case k = 0, it is obvious that µ(n) is disjoint from such e(f (n)) by the Prime Number Theorem. For the case k = 1, we have the following result. In fact, we have the following general result.
Proposition 1.6. Suppose that f (n) is a real-valued arithmetic function satisfying that the set {e(△f (n)) : n = 0, 1, . . .} has finitely many limit points, and For the case k ≥ 2, we obtain the following result.
Theorem 1.7. Let τ ∈ (5/8, 1) and k ≥ 2. Let f (n) be a real-valued arithmetic function satisfying when n large enough, for some positive constant C. Then The first ingredient of our proof of the above result is that if the k-th difference of f (n) decays to zero, then e(f (n)) can be approximated uniformly by certain concatenations of polynomial phases (see Lemma 4.1). The second one is Matomäki-Radziwi l l-Tao-Teräväinen-Ziegler's estimate [30] on averages of the correlation of multiplicative functions with polynomial phases in short intervals.
In the following, we provide a sufficient condition, which is weaker than SMDC (see Corollary 3.3), for Problem 1.
Proposition 1.8. Let k be a given positive integer. Denote D k by the set of all polynomials in R[y] of degrees less than k. Assume that the following estimate holds, Then for any f (n) satisfying lim n→∞ △ k f (n) = 0, we have Remark 1.9. All the results listed in this section also hold if µ is replaced by a more general "non-pretentious" 1-bounded (i.e., the l ∞ -norm is bounded by 1) multiplicative functions, such as the Liouville function and µ(n)χ(n), where χ is a given Dirichlet character.

Preliminaries on anqie entropy of arithmetic functions
To use tools from operator algebra to study Sarnak's conjecture, Ge introduced the anqie entropy for arithmetic functions [10]. 2.1. Some properties of anqie entropy. The anqie entropy has many nice properties. Here we list some properties which will be used in this paper. The following one is about the algebraic operations. Here and in the sequel, we use AE(f ) to denote the anqie entropy of any bounded arithmetic function f (n).
Proposition 2.1. For any bounded arithmetic functions f, g and continuous function φ(z) in C, we have The next one is about the lower semi-continuity of anqie entropy.
is a sequence of bounded arithmetic functions converging to f (n) uniformly with respect to n ∈ N, then lim inf N →∞ AE(f N ) ≥ AE(f ).
We refer readers to [10,Section 4] and [41] for details of the above two propositions. Arithmetic functions of zero anqie entropy can be realized in topological dynamical systems of zero topological entropy [10,Section 3]. Based on this fact, we have the following result.
Anqie entropy of arithmetic functions with finite ranges. In this subsection, we show some results on computing anqie entropy of functions with finite ranges, which will be used in the proofs of Theorems 1.2 and 1.4. Let us first recall some basic concepts in symbolic dynamical systems. For a finite set A, a block over A is a finite sequence of symbols from A. A J-block is a block of length J (J ≥ 1). For any given (finite or infinite) sequence x = (x 0 , x 1 , . . .) of symbols from A, we say that a block w occurs in x or x contains w if there are natural numbers i, j with i ≤ j such that (x i , . . . , x j ) = w. A concatenation of two blocks w 1 = (a 1 , . . . , a k ) and w 2 = (b 1 , . . . , b l ) over A is the block w 1 w 2 = (a 1 , . . . , a k , b 1 , . . . , b l ).
Now suppose that f : N → C has finite range. Let B J (f ) denote the set of all J-blocks occurring in f , i.e., The set of all regularly effective blocks in f is denoted by B e,r J (f ). For a function f taking finitely many values, as we mentioned previously, are regularly effective J-blocks in f . Then for each l > l J , there is at most one regular (lJ)-block in f which is not a concatenation of regularly effective J-blocks in f . This implies |B r lJ (f )| ≤ |B e,r J (f )| l + 1. Therefore Let k, m ≥ 1 be integers and c 1 , . . . , c m ∈ R be constants. Suppose that for j = 1, . . . , m, F j is a non-zero linear function of x 1 , ..., x k and H j is the hyperplane in R k given by F j (x 1 , ..., x k ) = c j . Denote by ., x k ) < c j }. A non-empty subset P of R k of the following form P = P 1 ∩ P 2 ∩ · · · ∩ P m , where each P j ∈ {H + j , H − j , H j }, is called a piece of R k cut by H 1 , ..., H m . In the following lemma we give an upper bound for the cardinality of pieces of R k cut by hyperplanes. In particular, W (m, k) ≤ (k + 1)2 k m k .
Proof. Notice that W (1, k) = 3 for any k ≥ 1. If we have then one can easily draw the conclusion by induction on m.
Now we show formula (2.7) holds for m ≥ 2. Let ψ be the map from the set of all pieces of R k cut by H 1 , ..., H m , denoted by P, onto the set of all pieces of R k cut by H 2 , ..., H m , denoted by P, given by ψ : Then for any piece D ∈ P, |ψ −1 (D)| ≤ 3. Now we show that if |ψ −1 (D)| ≥ 2, then H 1 ∩ D is nonempty. In fact, the only case we need to consider is when both H + 1 ∩ D and H − 1 ∩ D are nonempty. In this case, suppose does not exceed the cardinality of pieces of the form H 1 ∩ P 2 ∩ · · · ∩ P m in P times 2. In the following, we estimate this cardinality.
For each j = 2, . . . , m, H 1 ∩ H j is a hyperplane in H 1 , or empty, or equal to H 1 . Denote G 1 , ..., G n to be the ones which are hyperplanes in H 1 . Then n ≤ m − 1. We claim that if . . , G n . In fact, for j = 2, . . . , m, there are at most three cases. When . Therefore the inequality (2.7) holds and the proof is completed.

Proofs of Theorems 1.2 and 1.4
Recall that x = inf m∈Z |x − m| = min{{x}, 1 − {x}}. Then · defines a metric on R/Z and the topology induced by it on R/Z is equivalent to the Euclid topology on the unit circle. The following lemma will be used in this section.
for some k ∈ N. Then, for any ε > 0 and positive integer m ≥ 1, there is some L ∈ N such that, whenever n > L, the following holds for any j with 0 ≤ j ≤ m − 1, where Y n (n + j) is defined to be f (n + j) when 0 ≤ j ≤ k − 1 and to be the value determined by the following linear equations when k ≤ j ≤ m − 1, Proof. When k = 0, the claim is trivial. In the following, we assume k ≥ 1. We use induction on m to prove the lemma. For m ≤ k, since Y n (n + j) = f (n + j) for j = 0, 1, . . . , m − 1, choose L = 0. Then we obtain the claim in the lemma. Assume inductively that the claim holds for some m 0 ≥ k. In the following we shall prove the claim holds for m 0 + 1 case. By condition (3.1) and Proposition A.1, Then for any ǫ > 0, there is an L 1 > 0 such that whenever n > L 1 , By the induction hypothesis, there is an L 0 ∈ N, whenever n > L 0 , Define Y n (n + m 0 ) to be the value determined in the following equation, Then by equation (3.5), Combing with equation (3.4), we obtain the claim for m = m 0 +1, completing the induction.
Then for any real-valued arithmetic function f (n) satisfying for some k ∈ N, e(f (n)) and e(σ(f )(n)) are deterministic.
We use the following strategy to prove AE(f (n)) = 0. Firstly, we construct a sequence {g N (n)} ∞ N =0 of arithmetic functions with finite ranges to approach f (n) with respect to · R/Z . Then it suffices to prove AE(g N ) = 0 for N large enough by Proposition 2.2. Secondly, we decompose B e,r 2 m (g N ), the set of all 2 m -regularly effective blocks occurring in g N , into some subsets A ν,m according to the "zero entropy partition" given in g(n). Thirdly, for each ν, we construct a family of hyperplanes H ν,0 , . . . , H ν,2 m −1 in R q based on the approximation of f (n) to polynomials after differentiation. Then there is a correspondence between the set of pieces of R q cut by H ν,0 , ..., H ν,2 m −1 and A ν,m . Precisely, each element in A ν,m uniquely determines a piece of R q cut by H ν,0 , ..., H ν,2 m −1 . So |A ν,m |, moreover |B e,r 2 m (g N )|, is bounded by W (2 m , q), which has the polynomial growth rate with respect to 2 m by Lemma 2.6.
Proof of Theorem 1.2. In the following, we first prove e(f (n)) is deterministic. Then a similar argument leads to e(σ(f )(n)) deterministic since △ k f (n) = △ k+1 (σ(f ))(n) by the fact that σ is the inverse of △.
Given N ≥ 1, by Lemma 3.1, for each integer m ≥ 1, there is a sufficiently large L m ∈ N with 2 m |L m , such that whenever n ≥ L m , we have where Y n (n + j) is defined to be f (n + j) when 0 ≤ j ≤ k − 1 and the value determined by the following linear equations when k ≤ j ≤ 2 m − 1, Moreover, we may further assume that the sequence {L m } ∞ m=0 (L 0 = 0) chosen above satisfies L m+1 > L m for each m ≥ 1. Let d m = (L m+1 − L m )/2 m . Then the following is a partition of N, We define Y L 0 +a (L 0 + a) = f (L 0 + a) for a = 0, 1, . . . , L 1 − 1 and define the arithmetic function g N as follows: for a = 0, . . . , d m − 1 and j = 0, . . . , 2 m − 1, To prove e(f (n)) deterministic (that is AE e(f (n)) = 0 by Proposition 2.3), it suffices to prove AE(g N (n)) = 0 for N large enough by Propositions 2.1 and 2.2.
In the remaining part of this proof, we shall prove AE(g N ) = 0 for any given N ≥ 1. Let η be the function defined as η(n) = v if n ∈ S v , v = 1. . . . , w. Then the anqie entropy of η is zero.
Note that η(n) has finite range. By formula (2.2), Recall that B e,r 2 m (g N ) denotes the set of all 2 m -regularly effective blocks occurring in g N . Then In the following, we estimate the cardinality of A ν,m for each given m with 2 m > max(k + 1, d) and . Denote by q = k + wd. If q = 0 (i.e., k = 0 and d = 0), then by the definition, g N (n) = 0 when n ≥ L 1 . So it is easy to see that AE(g N ) = 0. Then AE(e(f )) = 0.
Next, we prove Theorem 1.4, which discusses about the anqie entropy of exponential functions of concatenations.
In the following, to show AE(g N ) = 0, we estimate |B e,r 2 m (g N )| for any given m with 2 m > max(k + 1, d), where |B e,r 2 m (g N )| is the cardinality of all regularly effective 2 m -blocks occurring in g N . Let i m be large enough such that N i+1 − N i > 2 m whenever i > i m . Let (g N (n2 m ), g N (n2 m + 1), . . . , g N (n2 m + 2 m − 1)) be a 2 m -block in g N with n2 m > max{L m , N im }. It is easy to see that there are two cases about this block: one case is that there is an i n such that (g N (n2 m ), g N (n2 m + 1), . . . , g N (n2 m + 2 m − 1)) = (g N,in (n2 m ), g N,in (n2 m + 1), . . . , g N,in (n2 m + 2 m − 1)); the other case is that there are two integers i n ≥ 0 and j n ≥ 1 such that g N (n2 m + j) = g N,in (n2 m + j) when j = 0, 1, . . . , j n − 1 and g N (n2 m + j) = g N,in+1 (n2 m + j) when j = j n , j n + 1, . . . , 2 m − 1. So |B e,r 2 m (g N )| is less than or equal to 2 m × |C m | 2 . Note that N, k, d, q are parameters independent of m. Then by formula (3.26), Now, we complete the proof of the theorem.
As an application of Theorem 1.4, we show that under the assumption of SMDC, for any deterministic sequence ξ(n), µ(n)ξ(n) does not correlate with e(f (n)) in short intervals on average when f (n) satisfies certain conditions. Precisely, Theorem 3.2. Let g(n) be defined as in Theorem 1.2. Suppose that D is a family of real-valued arithmetic functions such that holds for some k ≥ 1. Then SMDC implies that, for any deterministic sequence ξ(n), be a given sequence of natural numbers with N 0 = 0 and lim i→∞ ( If we take ξ(n) = 1 n≡a(mod q) (n) and D the set of all polynomials of degrees less than a given positive integer in Theorem 3.2, then we have the following corollary.  As observed in [37], equation (3.29) is implied by the local higher order Fourier uniformity conjecture, which is deduced from the Chowla conjecture ( [5], see also [34]). It is known that Chowla's conjecture implies SMDC. Corollary 3.3 shows that equation (3.29) can be deduced from SMDC.
Using a similar idea to the proof of Theorem 1.2, we obtain the following proposition that gives many characteristic functions with zero anqie entropy. Then 1 S (n), the characteristic function defined on S, is a deterministic sequence.
Proof. Assume that the degrees of p 1 (n) and p 2 (n) are both less than k. Then △ k p 1 (n) = △ k p 2 (n) = 0. In the following, for any given integer J ≥ k + 1, we estimate |B J (1 S )|, the cardinality of the set of all J-blocks occurring in 1 S .
(3.34) Moreover, the above inequalities also hold for each point in P . From formulas (3.33) and (3.34), it is not hard to see that for any given j with 0 ≤ j ≤ J − 1, As an application of the above proposition, we give the following example that satisfies the condition in Theorem 1.2 with g(n) = △ 2 f (n) = 0. Then 4. The Möbius disjointness of e(f (n)) with the k-th difference of f (n) tending to zero In this section, we shall study the disjointness of the Möbius function from exponential functions of arithmetic functions with the k-th differences tending to a constant (Propositions 1.5, 1.6, 1.8, and Theorem 1.7). We first show the following property that arithmetic functions with k-th differences tending to zero can be approximated by certain concatenations of polynomials of degrees less than k. where g N is the concatenation of {p i (n)} ∞ i=0 with respect to {N i } ∞ i=0 . Proof. By Lemma 3.1, for each integer m ≥ 1 and N ≥ 1, there is a sufficiently large L m ∈ N with 2 m |L m such that, whenever n ≥ L m , we have where Y n (n + j) is defined to be f (n + j) when 0 ≤ j ≤ k − 1 and the value determined by the following linear equations when k ≤ j ≤ 2 m − 1, It is not hard to check that We may further assume that the sequence {L m } ∞ m=0 (L 0 = 0) chosen above satisfies L m+1 > L m for each m. Let d m = (L m+1 − L m )/2 m . Then the following is a partition of N.
It is a polynomial of degree less than k. Let By formula (4.2) and equation (4.3), we obtain formula (4.1).
be an increasing sequence of natural numbers with N 0 = 0 and Proof. It is obvious that (4.5) ⇒ (4.6). We now show (4.6) ⇒ (4.5). In this process, we need to use a classical result (see [6] for k = 2 and [15, Chapter 6, Theorem 10] for k > 2) stated as follows, where the implied constant at most depends on k and q. By the above inequality and equation (4.6), for any given ǫ > 0, there is a positive integer M such that whenever m ≥ M and N ≥ N M , we have and sup i∈N 1≤n≤N n≡a(mod q) µ(n)e(p i (n)) < (ǫ/4)N. (4.8) n≡a(mod q) µ(n)e(p i (n)) + N l ≤n≤N n≡a(mod q) µ(n)e(p l (n)) <(ǫ/2)N l + (ǫ/2)N < ǫN.
We now prove Proposition 1.8, which gives a sufficient condition of disjointness between the Möbius function and exponential functions of arithmetic functions with the k-th differences tending to 0. Next, we shall prove Propositions 1.5, 1.6. It is easy to see that Proposition 1.5 directly follows from Proposition 1.6, so in the following, we just give the proof of Proposition 1.6. Before proving it, we need some preparations. The following Dirichlet's approximation theorem is classical and well-known (see e.g., [38,Section 8.2]), which is proved via the fact that if there are m + 1 points contained in m regions, then there must be at least two points lie in the same region. Lemma 4.3. Given L real numbers θ 1 , . . . , θ L and a positive integer q, then we can find an integer t ∈ [1, q L ], and integers a 1 , . . . , a L such that |tθ j − a j | ≤ 1/q, j = 1, 2, . . . , L.
The following "asymptotical periodicity" of the concatenation of certain linear phases will be used in the proof of Proposition 1.6.
be an increasing sequence of integers with N 0 = 0 and lim i→∞ (N i+1 − N i ) = ∞. Suppose that α 0 , α 1 , . . . are real numbers such that the sequence {e(α i )} ∞ i=0 has finitely many limit points. Assume that f (n) = e(nα i )e(β i ) when N i ≤ n < N i+1 for i = 0, 1, 2, . . ., where β 0 , β 1 , . . . are real numbers. Then there is a δ with 0 < δ < 1 and a sequence {n j } ∞ j=0 of positive integers with lim j→∞ n j = ∞ such that Proof. Suppose that the limit points of {e(α i )} ∞ i=0 are e(θ 1 ), . . . , e(θ L ) for some integer L ≥ 1. Let q j = (j + 6) 2 π 2 , j ≥ 0. By Lemma 4.3, we can find an integer n j with 1 ≤ n j ≤ q L j such that |θ s − a s,j n j | ≤ 1 n j q j , where a s,j is some integer for s = 1, . . . , L. It is not hard to check that the sequence {n j } ∞ j=0 can be chosen to satisfy lim j→∞ n j = ∞. Moreover, there is an i 0 such that when i ≥ i 0 we can choose an s ∈ {1, 2, . . . , L} satisfying α i − θ s < 1 n j q j . Then for i ≥ i 0 , n j α i < 2 q j and |e(n j α i ) − 1| = 2| sin(πn j α i )| ≪ 1 q j . So, for any given n j , The claimed result follows by choosing δ = 1 2L .
To prove Proposition 1.6, we also need the following result of the second author [43, Lemma 4.1] on the self-correlation of µ(n)e(P (n)) in short arithmetic progressions. where ϕ is the Euler totient function and the implied constant depends on d at most.
We are left to prove claim (4.11). Write as a sequence of real numbers such that Define g(n) to be e(nα i )e(β i ) when N i ≤ n < N i+1 , i = 0, 1, . . .. So it suffices to prove that (4.14) Note that the right side of the above inequality tends to zero as j → ∞. Hence for any given ǫ > 0, by formulas (4.13) and (4.14), there are positive integers j 0 and M 0 such that whenever N > M 0 , Then by the Cauchy-Schwarz inequality, µ(n + ln j 0 )e f 0 (n + ln j 0 ) g(n + ln j 0 ) + O( ln j 0 N ).
In the remaining part of this section, we shall prove Theorem 1.7. The major ingredient of our proof is Matomäki-Radziwi l l-Tao-Teräväinen-Ziegler's recent work [30] on averages of the correlation between multiplicative functions and polynomial phases in short intervals. To state this result, we shall use the following distance function of Granville and Soundararajan, for any multiplicative function g(n) with |g(n)| ≤ 1 for all n ∈ N. This distance function was introduced in [12] to measure the pretentiousness between g(n) and n → n it . Throughout this section, define Let f : N → C be a multiplicative function with |f (n)| ≤ 1 for any n ∈ N. Suppose that X ≥ 1, X θ ≥ H ≥ exp((log X) τ ), and η > 0 are such that Then M(f ; AX k /H k−1/2 , Q) ≪ k,η,θ,τ 1 for some positive numbers A, Q ≪ k,η,θ,τ 1.
The following is a well-known result (see, e.g., [29, (1.12)]) about the "non-pretentious" nature of the Möbius function. By the above proposition and Lemma 4.6, we have the following result which states that µ(n) does not correlate with polynomial phases in short intervals on average. Lemma 4.8. Let k be a given positive integer, and let 5/8 < τ < 1 and 0 < θ < 1 be fixed. Suppose that X ≥ 1 and X θ ≥ H ≥ exp((log X) τ ). Denote by D k the set of all polynomials in R[y] of degrees less than k. Then for any η > 0, there is an X 0 > 0 (at most depends on k, η, θ, τ ) such that whenever X > X 0 ,  where D k is the set of polynomials in R[y] of degrees less than k. Let S j = {lh m + j : l = 0, 1, · · ·, ⌊ Nm hm ⌋}, for 0 ≤ j ≤ h m − 1. Then, by (4.19), there is a j 0 such that (4.20) By formula (4.20), for N m sufficiently large. Inserting this into the above inequality and letting m → ∞, we obtain lim sup Letting ǫ → 0, we obtain equation (4.18).
Now we are ready to prove Theorem 1.7, which states the Möbius disjointness of e(f (n)) with the k-th difference of f (n) tending to zero as in formula (1.5).
The following simple fact is used in this paper.
where {·} denotes the fractional part function.
The following gives an estimate of f (n) through the initial values and the upper bound of △ k f (n).

Appendix B. A lemma
In this section, we prove the following lemma which is used in the proof of Theorem 3.2. There are some other methods to prove the following result, while for self-containing, we provide a proof that is adapt to our situation. Lemma B.1. Let w(n) be a bounded arithmetic function and D be a non-empty family consisting of real-valued arithmetic functions. Then the following two conditions are equivalent. Proof. We first prove that (i)⇒(ii). Assume on the contrary that formula (B.2) does not hold. Then there is a δ ∈ (0, 1) and a sequence {h j } ∞ j=0 of positive integers with δh j > 4 and lim j→∞ h j = ∞, such that lim sup Given h j , choose X j large enough with δX j > 16h j and X j > 4X j−1 satisfying 2X j X j sup f ∈D x≤n<x+h j w(n)e(f (n)) dx > 2δX j h j .