Averaging with the Divisor Function: ℓ p -improving and Sparse Bounds
aa r X i v : . [ m a t h . C A ] F e b AVERAGING WITH THE DIVISOR FUNCTION: ℓ p -IMPROVING ANDSPARSE BOUNDS CHRISTINA GIANNITSI
Abstract.
We study averages along the integers using the divisor function d ( n ), and definedas K N f ( x ) = 1 D ( N ) X n ≤ N d ( n ) f ( x + n ) , where D ( N ) = P Nn =1 d ( n ). We shall show that these averages satisfy a uniform, scale free ℓ p -improving estimate for p ∈ (1 , (cid:18) N X | K N f | p ′ (cid:19) /p ′ . (cid:18) N X | f | p (cid:19) /p as long as f is supported on [0 , N ].We will also show that the associated maximal function K ∗ f = sup N | K N f | satisfies ( p, p )sparse founds for p ∈ (1 , K ∗ is bounded on ℓ p ( w ) for p ∈ (1 , ∞ ), forall weights w in the Muckenhoupt A p class. Contents
1. Introduction 12. Preliminaries 43. Approximating Multipliers 74. Fixed Scale 95. Sparse Bounds 11References 141.
Introduction
We establish ℓ p -improving and sparse bounds for the discrete averages formed from thedivisor function. Let d ( n ) be the usual divisor function, D ( N ) = P Nn =1 d ( n ), and consider Research supported in part by grant from the US National Science Foundation, DMS-1949206. the following averaging operator K N f = 1 D ( N ) X n ≤ N d ( n ) f ( x + n )and the associated maximal function K ∗ f = sup N | K N f | . Let us first establish some notation we shall be using throughout the paper. For twoquantities a and b , we shall write a . b if there exists a positive constant C such that a ≤ C b .We shall write a . p b if they implied constant depends on p . For a function f on the integers,and an interval I ⊂ Z , we write h f i I,p = | I | X x ∈ I | f ( x ) | p ! /p Also for an interval I = [ a, b ] ∩ Z we will write 2 I = [2 a − b − , b ] ∩ Z , 3 I = [2 a − b − , b − a +1] ∩ Z for the doubled and tripled interval respectively. We shall also use ˆ f or F f for the Fouriertransform of f , defined as F f ( θ ) = X x ∈ Z f ( k ) e − πixθ , and ˇ f or F − for the inverse Fourier transform, defined as F − f ( x ) = Z ˆ f ( θ ) e πixθ dθ. Finally, if 1 ≤ p ≤ ∞ , let p ′ denote its conjugate exponent, that is 1 p + 1 p ′ = 1 . Our two main results are a scale free, ℓ p improving inequality, and a sparse bound for themaximal function K ∗ . More specifically, Theorem 1.1.
For p ∈ (1 , ∞ ) , there exists C p > such that for all positive integers N andfunctions f supported on an interval E of length N , there holds (1.2) h K N f i E,p ′ ≤ C p h f i E,p
Now a collection S of intervals is called sparse if for every I ∈ S there exists a set E I ⊂ I so that | E I | > | I | / E I for I ∈ S are pairwise disjoint. VERAGING WITH THE DIVISOR FUNCTION: ℓ p -IMPROVING AND SPARSE BOUNDS 3 Theorem 1.3.
For r, s ∈ (1 , , the maximal operator K is of ( r, s ) -sparse type, that is thereexists C > such that for all compactly supported functions f and g , there exists a sparsecollection S so that the ℓ ( Z ) inner product satisfies (1.4) | ( K ∗ f, g ) | ≤ C X E ∈S | I |h f i I,r h g i I,s
The sparse result of Theorem 1.3, implies not only ℓ p boundedness for the maximal opera-tor, but also weighted inequalities for weights w for which the ordinary maximal function isbounded, which is a new result. Specifically, Corollary 1.5.
For any p ∈ (1 , ∞ ) and all weights w in the Muckenhoupt class A p , themaximal operator K ∗ : ℓ p ( w ) → ℓ p ( w ) is a bounded operator. There is a plethora of papers focusing on the subject of sparse bounds and ℓ p improvinginequalities, starting with [CKL19, Hug20], and [KL18]. It was followed by [Kes18a, Kes18b,KLM19, KLM20]. More recent results include [HLY19, HKLY20, HKL +
19] and [DDL20]. Weshould also mention the work in [CW14], where they study ergodic theorems using the divisorfunction.Our approach is inspired by [HKLY20] and [HLY19]. To prove the estimates, we usethe same two techniques as in the aforementioned papers, adjusted to fit our particularsetting. Particularly, we decompose our operator into a High pass and a Low pass term[HKLY20, HLY19, Ion04, KLM20]. This decomposition is key to proving both the ℓ p improv-ing and the sparse result, although the sparse decomposition is somewhat different. In bothcases, Ramanujan sums have an important role to play in estimating the low pass term, and sodo results from the number-theoretic literature, involving exponential sums using the divisorfunction, [Jut84]. Acknowledgment.
The author would like to thank Michael Lacey for his continuousguidance, support, and encouragement.
CHRISTINA GIANNITSI Preliminaries
Recall the major and minor arcs decomposition. Let A Q = { ≤ A < Q : (
A, Q ) = 1 } denote the multiplicative group associated to an integer Q >
1. For s ≥ R s = (cid:26) AQ ∈ [0 ,
1) : A ∈ A Q , s − ≤ Q < s (cid:27) For 0 < ε ≤ / AQ ∈ R s , with s ≤ jε , we define the j-th major arc at A/Q as M j (cid:18) AQ (cid:19) = (cid:26) AQ + η : AQ ∈ R s , | η | < ( ε − j (cid:27) Those are disjoint for ε small enough. The j -th major arcs are given by M j = S AQ ∈R s M j ( A/Q ).We define the j -th minor arcs m j as the complement of M j .Also recall the Ramanujan sums, defined as(2.1) c Q ( n ) = X A ∈ A Q e ( An/Q ) , where e ( x ) := e πix .We present the following lemma, based on an important property of Ramanujan sums whichis due to Bourgain [Bou93]. An alternate proof can be found in [KLM19, Lemma 3.13]. Ourproof is fairly simple and follows the same steps as [HKLY20, Lemma 3.4]. Lemma 2.2.
For any ε > and integer k > , uniformly in N > J k , there holds (2.3) N X | x | We give an outline of the proof. Bourgain’s estimate [Bou93, (3.43)], tells us that forany integer J N X | x | Let us now turn our attention to the divisor function. The asymptotics of the divisorfunction are already well known. If N, n > d ( n ) is the divisor function then(2.4) log d ( n ) . log n log log n = o (log n )(2.5) D ( N ) = N log N + (2 γ − N + O ( √ N ) = O ( N log N )where γ is Euler’s constant.The kernel of our operator has been extensively studied in the number theory literature. Weshall be using results discussed in [Jut84] and [CW14]. Particularly, the following is Lemma5.4 from [CW14], for P N = N / and Q N = N − ε . We have also normalized the estimate bydividing by D ( N ). Lemma 2.6. Suppose N ≥ and < ε ≪ . Let θ ∈ [0 , be such that for every ≤ Q ≤ √ N and every A ∈ A Q , | θ − A/Q | > N − ε . Then the following is true. (2.7) 1 D ( N ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ≤ n ≤ N d ( n ) e ( nθ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O ( N − ε )Now letˆΓ N ( θ ) = 1 N N − X n =0 ˆ δ n ( θ ) = 1 N N − X n =0 e ( nθ )denote the Fourier transform usual averages. It is a well known fact that these averages satisfy(2.8) | ˆΓ N ( θ ) | . min (cid:26) , N | θ | (cid:27) for | θ | < / 2. With that notation we have the following result. CHRISTINA GIANNITSI Lemma 2.9. Suppose N ≥ , Q ≤ √ N and < ε ≪ . Let γ denote Euler’s constant.There exists a constant C such that for all A ∈ A Q , (2.10a) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n ≤ N d ( n ) e ( nA/Q ) − N (log N − Q + 2 γ − Q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C N + ε There exists a constant C , such that if < | η | < Q √ N , then for all A ∈ A Q , (2.10b) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n ≤ N d ( n ) e (cid:16) n (cid:0) AQ + η (cid:1)(cid:17) − e ( η ) − πiη · N (log N − Q + 2 γ − 1) ˆΓ N ( η ) Q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C N + ε There exists a constant C , such that if < | θ | < Q √ N , then (2.10c) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n ≤ N d ( n ) e ( nθ ) − e ( θ ) − πiθ · NQ (log N + 2 γ − 1) ˆΓ N ( θ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C N + ε Lemma 2.9 is a direct result of Theorem 5 of [Jut84], along with with Remarks 1, 2, andequation (1.4) of the same paper, as well as the fact that N − X n =0 e ( − nη ) = e ( N η ) − e ( η ) − . Remark . We point out that Lemma 2.9 is an estimate of the sum before normalization.Therefore, using (2.5), the implied estimate for the differences is of the order N − + ε . Remark . If we look at the Taylor expansion of e ( θ ) − πiθ , we can easily see that (cid:12)(cid:12)(cid:12)(cid:12) e ( θ ) − πiθ − (cid:12)(cid:12)(cid:12)(cid:12) . | θ | . Remark . Note that by (2.5) we have that | N (log N + 2 γ − − D ( N ) | . √ N . VERAGING WITH THE DIVISOR FUNCTION: ℓ p -IMPROVING AND SPARSE BOUNDS 7 Approximating Multipliers We consider a Schwartz function χ that satisfies [ − , ] ≤ χ ≤ [ − , ] , and its dilations χ s ( θ ) = χ (2 s θ ). Now consider the following multipliers, for s ≥ s ≤ Q < s +1 .(3.1a) ˆ L ,N ( θ ) = 2 ˆΓ N ( θ ) χ ( θ )(3.1b) ˆ L Q,N ( θ ) = 2 Q X A ∈ A Q ˆΓ N ( θ − A/Q ) χ s ( θ − A/Q )Using the definition of Ramanujan sums c Q ( x ) in (2.1), we calculate the inverse Fouriertransform of our multipliers. Lemma 3.2. With the definition of (3.1) we have (3.3a) L ,N ( x ) = 2 Γ N ∗ ˇ χ (3.3b) L Q,N ( x ) = 2 Q c Q ( x ) Γ N ∗ ˇ χ s Proof. We compute L ,N ( x ) = 2 Z T ˆΓ N ( θ ) χ ( θ ) e ( xθ ) dθ = 2Γ N ∗ ˇ χ L Q,N ( x ) = 2 Q X A ∈ A Q Z T ˆΓ N ( θ − A/Q ) χ s ( θ − A/Q ) e ( χθ ) dθ = 2 Q X A ∈ A Q e ( xA/Q ) Z T ˆΓ N ( θ ) χ s ( θ ) e ( χθ ) dθ = 2 Q X A ∈ A Q e ( xA/Q ) Γ N ∗ ˇ χ s = 2 Q c Q ( x ) Γ N ∗ ˇ χ s (cid:3) Theorem 3.4. Let N > and < ε < / . If < P ≤ √ N then (3.5) ˆ K N = P X Q =1 ˆ L Q,N + ˆ r N,P , where k ˆ r N,P k ∞ . P − ε . CHRISTINA GIANNITSI Proof. The first thing we need to note is that, by construction, our multipliers ˆ L Q,N aresupported on disjoint intervals, centered at rationals A/Q , for 2 s ≤ Q < s +1 , s ≥ 1, and A ∈ A Q . Using (2.8), we have that(3.6) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X s ≤ Q< s +1 ˆ L Q,N (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ . − s We first show the theorem holds when P = √ N . Let us fix ε > θ ∈ T . We distinguishthree cases. First Case. Suppose there exists Q ≤ √ N , such that for some A ∈ A Q , θ satisfies | θ − A/Q | < N − ε . Then for any other B ∈ A Q with B = A , we have χ s ( θ − B/Q ) = 0, while χ s ( θ − A/Q ) = 1. This means that all the other terms in the sum of our approximatingmultipliers vanish, and we are only left withˆ L Q,N ( θ ) = 2 Q ˆΓ N ( θ − A/Q )Notice that using (2.8), and Remarks 2.11, 2.12 and 2.13, we see that (cid:12)(cid:12)(cid:12) ˆ L Q,N ( θ ) − e ( θ − A/Q ) − πi ( θ − A/Q ) · N (log N − Q + 2 γ − Q · D ( N ) ˆΓ N ( θ − A/Q ) (cid:12)(cid:12)(cid:12)(cid:12) . | θ − A/Q | + 2 2 log Q + √ NQ D ( N ) . N − / Using this estimate, part (b) of Lemma 2.9 and the triangle inequality,(3.7) | ˆ K N ( θ ) − ˆ L Q,N ( θ ) | . N − / . For the rest of the Q ′ ≤ √ N , Q ′ = Q , we have that for all A ′ ∈ A Q ′ , | θ − A ′ /Q ′ | > N − ε , sousing (2.8) we get (cid:12)(cid:12)(cid:12) ˆ L Q,N ( θ ) (cid:12)(cid:12)(cid:12) . N − ε Q ′ Summing over all such Q ′ , we get(3.8) X ≤ Q ′ ≤√ NQ ′ = Q (cid:12)(cid:12)(cid:12) ˆ L Q ′ ,N ( θ ) (cid:12)(cid:12)(cid:12) . N − + ε VERAGING WITH THE DIVISOR FUNCTION: ℓ p -IMPROVING AND SPARSE BOUNDS 9 Using (3.6), (3.7), and (3.8) we can get the wanted bound for this choice of θ . It is worthmentioning that the value of ε might change from one line to the next, but as always it denotesa small positive number. Second Case. This is the case where | θ | ≤ N − ε . This case works exactly like the previousone, except we use part (a) of the definition of the multipliers, equation (3.1), and parts (a)and (c) of the estimate in Lemma 2.9. Third Case. Our last case is when θ does not meet any of the criteria of the previous twocases. This estimate is similar to our estimate of (3.8). Indeed, using (2.8), and the fact thatnow | θ − A/Q | > N − ε for all choices of Q ≤ √ N and A ∈ A Q , we get that(3.9) X ≤ Q ≤√ N (cid:12)(cid:12)(cid:12) ˆ L Q ′ ,N ( θ ) (cid:12)(cid:12)(cid:12) . X ≤ Q ≤√ N N − ε Q ′ . N − + ε This, combined with (2.7) concludes this case. This implies that ˆ K N = P √ NQ =1 ˆ L Q,N + ˆ r N, √ N , where k ˆ r N, √ N k ∞ . ( √ N ) − ε .Now suppose that P < √ N . Then we see that ˆ K N = P PQ =1 ˆ L Q,N + ˆ r N,P , whereˆ r N,P = √ N X Q = P +1 ˆ L Q,N + ˆ r N, √ N Using the same argument as in (3.6), we see that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) √ N X Q = P +1 ˆ L Q,N (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ ≤ max P +1 ≤ Q ≤ N | ˆ L Q,N | ≤ P + 1 . P − ε . This, combined with the fact that k ˆ r N, √ N k ∞ ≤ ( √ N ) − ε . P − ε completes the proof. (cid:3) Fixed Scale We will now focus on a fixed scale estimate, so let us fix an N , let E be an interval suchthat | E | = N and fix p ∈ (1 , f = F supported on E and g = G supported on E then(4.1) 1 N ( K N f, g ) ≤ C p h f i E,p h g i E,p , where C p is positive constant independent of N . The proof of Theorem 1.1 follows immediatelyby an additional elementary argument.Notice that it is trivial to obtain the following bound using (2.4):1 N ( K N f, g ) ≤ N δ h f i E, h g i E, , for a fixed δ > 0. This immediately implies that if2 N δ h f i /p ′ E, h g i /p ′ E, ≤ h f i E, ≥ − p ′ N − δ p ′ h g i E, ≥ − p ′ N − δ p ′ . (4.2)We shall show that (4.1) still holds by using an auxiliary result, the high-low decomposition.The idea for this decomposition goes back to Bourgain in [Bou99]. Lemma 4.3. Let p ∈ (1 , . Then there exists N p > such that for all N > N p and ≤ m ≤ √ N we can decompose K N f = H m + L m where h H m i E, . m − /p h f i / E, h L m i E, ∞ . m /p ′ h f i /pE, Remark . We can explicitly define N p as the smallest positive integer that satisfies(4.5) 2 < N (cid:16) − p (cid:17) − δ p (cid:16) − p (cid:17) p . Using this lemma we get that1 N ( K N f, g ) . m − p ′ ( h f i E, h g i E, ) / + m /p ′ h f i /pE, h g i E, Optimizing over m , that is m ∼ h f i − p E, h g i − E, , which is an allowed choice within range thanksto (4.2) and (4.5), and plugging this back to our estimate completes the proof of (4.1).Thus all we are missing is the proof of the lemma. VERAGING WITH THE DIVISOR FUNCTION: ℓ p -IMPROVING AND SPARSE BOUNDS 11 Proof of lemma 4.3. We shall decompose our operator using the decomposition of Theorem3.4 for P = m and ε = 1 /p . Then set H m = F − (ˆ r N,m ˆ f ), as in (3.5). The desired ℓ -estimatefollows immediately using Theorem 3.4.It is now obvious that L m has to be defined as the remaining approximating multipliers L m = F − (cid:16) ˆ K N − ˆ H m (cid:17) . Note that from Young’s convolution inequality we get | Γ N ∗ ˇ χ s | . N − . Using Remark 2.13, we can therefore compute | L m ( x ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m X Q =1 L Q,N ∗ f ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . m X Q =1 N X y =1 | c Q ( y ) | Q | Γ N ∗ ˇ χ s ( y ) | f ( x − y ) . N m X Q =1 2 N X y =1 | c Q ( y ) | Q f ( x − y ) . N N X y =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m X Q =1 | c Q ( y ) | Q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ′ /p ′ N N X y =1 f ( x − y ) /p by H¨older’s Inequality . m /p ′ h f i /pE, The last inequality comes from Lemma 2.2 with M = 2 N , J = m , k = p ′ , and ε = 1 /p ′ . Notethat this estimate requires N > N p . (cid:3) Sparse Bounds Let us now turn our attention to the ( p, p )-sparse bound for p ∈ (1 , Suppose E is an interval of length 2 n . Let f = F be supported on E and g = G besupported on G ⊆ E . Now, consider a choice of stopping time τ : E → { n : 1 ≤ n ≤ n } . Our goal is to prove an estimate for a particular type of stopping times, the definition ofwhich can also be found in [HKLY20]. Definition 5.1. A stopping time τ is admissible if and only if for any interval I ⊆ E suchthat h f i I, > h f i E, , there holds inf { τ ( x ) : x ∈ I } > | I | . Lemma 5.2. For all admissible stopping times, and for all < p < we have that (5.3) ( K τ f, g ) . ( h f i E, h g i E, ) /p | E | The proof of Theorem 5.2 follows the idea of [HKLY20]. We will again use the auxiliaryhigh-low construction, slightly modified this time. Once we have the individual estimates,the final result can be obtained using the same argument as in the previous section, and istherefore omitted. For integers M = 2 m , we decompose K τ f ≤ H + L where h H i E, . M − p h f i / E, (5.4) h L i E, ∞ . M p ′ h f i /pE, (5.5)In this case, the trivial bound is given by h K τ f i E, ∞ . sup x τ δ ( x ) · h f i E, . for some δ > 0. We pick δ so that 2( p ′ + 1) δ ≪ / 2. It is then obvious that we need only studythe operator on a set D = { x : τ δ ( x ) ≥ C p · M /p ′ } , where C p is a constant that dependsonly on p . Outside our set, the result holds as a consequence of the trivial bound.We need to know what happens to K ⌊ n /δ ⌋ f for n > C p M /p ′ . For convenience, let ℓ = n δp ′ /ε , where ε = 2( p ′ + 1) δ ≪ δ . Therefore, ℓ ≤ √ n , and by Theorem3.4 we getˆ K ⌊ n /δ ⌋ = X ≤ Q ≤⌊ ℓ /δ ⌋ ˆ L Q, ⌊ n /δ ⌋ + ˆ r ⌊ n /δ ⌋ , ⌊ ℓ /δ ⌋ with N = ⌊ n /δ ⌋ , P = ⌊ ℓ /δ ⌋ , and ε = ε . This means that k ˆ r ⌊ n /δ ⌋ , ⌊ ℓ /δ ⌋ k ∞ . ℓ − ε /δ . n − p ′ ,by the way we chose ℓ . VERAGING WITH THE DIVISOR FUNCTION: ℓ p -IMPROVING AND SPARSE BOUNDS 13 The high-low decomposition for this argument is a bit more involved. Particularly, when wedecompose the operator into the approximating multipliers and the remainder, the remaindergoes to the high pass term, as it did in the previous section. However, in this case we needto break the approximating multipliers further, into two pieces, one of which will become thelow-pass term, and the other will be added in the high-pass term. That is, our decompositionfor the operator now becomes K τ ≤ H + H + L , where now H = H + H .So we let H = | r ⌊ n /δ ⌋ , ⌊ ℓ /δ ⌋ ∗ f | . In that case h H i E, ≤ X n ≥ C p M /p ′ h r ⌊ n /δ ⌋ , ⌊ ℓ /δ ⌋ ∗ f i E, . X n ≥ C p M /p ′ n − p ′ h f i E, . M − p ′ h f i E, using a square function argument. The second contribution to the high-pass term is thefollowing H = sup n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X m 1, 2 s ≤ Q < s +1 as in the definition of (3.1).Now let us take a closer look at Γ τ ∗ ˇ χ s . Using (2.8) we can see that for | x | ≤ τ | Γ τ ∗ ˇ χ s ( x ) | . τ . Next, suppose 2 k τ < | x | ≤ k +1 τ , for k ≥ 2, and recall that χ s is a dilation of a Schwartzfunction, where s ≥ s ≤ Q < s +1 . Using (2.8) and the fact that M ≤ C p M δp ′ . τ ,we obtain | Γ τ ∗ ˇ χ s ( x ) | . X y ≤ τ | Γ τ ( y ) | | ˇ χ s ( x − y ) | . X y ≤ τ τ s (cid:12)(cid:12)(cid:12)(cid:12) ˇ χ (cid:18) x − y s (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) . s τ s k τ . M k τ . k τ Using these estimates in (5.6) we get | L ( x ) | ≤ τ X | y |≤ τ M X Q =1 | c Q ( y ) | Q ∗ f ( x − y ) + ∞ X k =2 k τ k +1 X | y | =2 k M X Q =1 | c Q ( y ) | Q ∗ f ( x − y ) . τ τ X y =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M X Q =1 c Q ( y ) Q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ′ p ′ τ τ X y =1 f ( x − y ) p ++ ∞ X k =2 k k +1 τ X y =2 k k +1 τ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M X Q =1 | c Q ( y ) | Q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ′ /p ′ k +1 τ k +1 τ X y =2 k f ( x − y ) p . M /p ′ h f i /pE, Here, we have used H¨older’s p − p ′ inequality and Lemma 2.2. Admissibility of τ was essentialto obtain the estimate in terms of the average of f . Our assumption that τ δ ( x ) > C p J /p ′ is also unavoidable, if we wish to use Lemma 2.2. Thus the constant C p has to be chosenaccordingly, and our proof is complete. References [Bou89] Jean Bourgain. Pointwise ergodic theorems for arithmetic sets. Publications Math´ematiques del’IH ´ES , 69:5–41, 1989.[Bou93] J. Bourgain. Fourier transform restriction phenomena for certain lattice subsets and applications tonon-linear evolution equations. Part I Schr¨odinger equations. Geom. Funct. Anal , 3, No. 2:107–156,1993. VERAGING WITH THE DIVISOR FUNCTION: ℓ p -IMPROVING AND SPARSE BOUNDS 15 [Bou99] Jean Bourgain. New global well-posedness results for nonlinear Schr¨odinger equations. AMS Publi-cations , 1999.[CKL19] Amalia Culiuc, Robert Kesler, and Micheal Lacey. Sparse bounds for the discrete cubic Hilberttransform. Anal. PDE , 12:1259–1272, 2019.[CW14] Christophe Cuny and Michel Weber. Ergodic theorems with arithmetical weights. Israel Journal ofMathematics , 217, 2014.[DDL20] Shival Dasu, Ciprian Demeter, and Bartosz Langowski. Sharp ℓ p -improving estimates for the discreteparaboloid. arXiv:2002.11758 , 2020.[HKL + 19] Rui Han, Vjekoslav Kovaˇc, Micheal Lacey, Jos`e Madrid, and Fan Yang. Improving estimates fordiscrete polynomial averages. arXiv:1910.14630 , 2019.[HKLY20] Rui Han, Ben Krause, Michael T. Lacey, and Fan Yang. Averages along the primes: Improving andsparse bounds. Concrete Operators , 7(1):45 – 54, 01 Jan. 2020.[HLY19] Rui Han, Micheal Lacey, and Fan Yang. Averages along the square integers: ℓ p improving andsparse inequalities. arXiv:1907.05734 , 2019.[Hug20] K. Hughes. ℓ p p-improving for discrete spherical averages. Annales Henri Lebesgue , 3:959–980, 2020.[Ion04] Alexandru D. Ionescu. An endpoint astimate for discrete spherical maximal function. Proc. Aper.Math. Soc. , 19:1411–1417, 2004.[Jut84] Matti Jutila. On exponential sums involving the divisor function. Journal fur die reine und ange-wandte Mathematik , 355:173–190, 1984.[Kes18a] Robert Kessler. ℓ p ( Z d )-improving properties and sparse bounds for discrete spherical maximal av-erages. arXiv:1805.09925 , 2018.[Kes18b] Robert Kessler. ℓ p ( Z d )-improving properties and sparse bounds for discrete spherical maximalmeans, revisited. arXiv:1809.06468 , 2018.[KL18] Robert Kesler and Micheal Lacey. ℓ p -improving inequalities for discrete spherical averages. arXiv:1804.09845 , 2018.[KLM19] Robert Kesler, Micheal Lacey, and Dario Mena. Lacunary discrete spherical maximal functions. New York Journal of Mathematics , 25:541–557, 2019.[KLM20] Robert Kesler, Micheal Lacey, and Dario Mena. Sparse bounds for the discrete spherical maximalfunctions. Pure Appl. Anal. 2 , no.1:75–92, 2020.[Kra18] Ben Krause. Discrete analogoues in harmonic analysis: Maximally monomially modulated singularintegrals related to Carleson’s theorem. arXiv:1803.09431 , 2018. Department of Mathematics, Georgia Institute of Technology, Atlanta GA 30332, USA Email address ::