On Large Values of Weyl Sums
Changhao Chen, Igor E. Shparlinski

TL;DR
This paper investigates the size and structure of exceptional coefficient sets where Weyl sums are unusually large, revealing they are quite large in terms of Baire category and Hausdorff dimension, especially for polynomials of degree d.
Contribution
It demonstrates that the sets of coefficients with large Weyl sums are large in Baire category and Hausdorff dimension, and extends results to sums with a single monomial.
Findings
Exceptional sets have positive Hausdorff dimension.
Sets are large in Baire category and Hausdorff sense.
Results apply to polynomials with poorly distributed fractional parts.
Abstract
A special case of the Menshov--Rademacher theorem implies for almost all polynomials of degree for the Weyl sums satisfy the upper bound Here we investigate the exceptional sets of coefficients with large values of Weyl sums for infinitely many , and show that in terms of the Baire categories and Hausdorff dimension they are quite massive, in particular of positive Hausdorff dimension in any fixed cube inside of . We also use a different technique to give similar results for sums with just one monomial . We apply these results to show that the set of poorly distributed modulo one polynomials is rather massive as well.
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On Large Values of Weyl Sums
Changhao Chen
Department of Pure Mathematics, University of New South Wales, Sydney, NSW 2052, Australia
and
Igor E. Shparlinski
Department of Pure Mathematics, University of New South Wales, Sydney, NSW 2052, Australia
Abstract.
A special case of the Menshov–Rademacher theorem implies for almost all polynomials of degree for the Weyl sums satisfy the upper bound
[TABLE]
Here we investigate the exceptional sets of coefficients with large values of Weyl sums for infinitely many , and show that in terms of the Baire categories and Hausdorff dimension they are quite massive, in particular of positive Hausdorff dimension in any fixed cube inside of . We also use a different technique to give similar results for sums with just one monomial . We apply these results to show that the set of poorly distributed modulo one polynomials is rather massive as well.
Key words and phrases:
Weyl sum, exceptional set, Vinogradov mean value theorem, rational exponential sums, Baire category, Hausdorff dimension
2010 Mathematics Subject Classification:
11K38, 11L15, 28A78, 28A80
Contents
1. Introduction
1.1. Motivation
Here we consider a new type of problems of metric number theory where the vectors of real numbers are classified by the size of the corresponding Weyl sums given by (1.1) below, rather than by their Diophantine approximation properties as in the classical settings, see [3, 8].
Clearly both points of view are ultimately related and operated in similar notions such as the Lebesgue measure and Hausdorff dimension. They are also both related to the question of uniformity of distribution modulo one of fractional parts of real polynomials. However, our study of sets of large Weyl sums also uses several new ideas and technics. We believe that these ideas and concrete results on such a very powerful and versatile tool as exponential sums can find applications to other problems. In particular, in Section 1.4 below we give one of such applications and show that the set of polynomials which are poorly distributed modulo one is rather massive (in fact, our results are quantitative and thus more precise).
In problems of this kind, the case is much harder than the case . The main reason is that Lemma 2.3 below, giving an exact size of Gauss sums, which we have for the case , does not in general have any analogues for , see also Remark 2.8 below.
1.2. Set-up and background
We now describe our main objects of study.
For an integer , let denote the -dimensional unit torus.
For a vector and , we consider the exponential sums
[TABLE]
which are commonly called Weyl sums, where throughout the paper we denote . From the Parseval identity
[TABLE]
one immediately concludes that for any fixed the set of with is of Lebesgue measure at most , which is nontrivial when .
Furthermore, from the Vinogradov mean value theorem, in the currently known form
[TABLE]
where , due to Bourgain, Demeter and Guth [2] (for ) and Wooley [22] (for ) (see also a more general form due to Wooley [24]), one can derive a much stronger bound when .
In fact, a special case of the Menshov–Rademacher theorem, see [10, p. 251], implies that for almost all (with respect to the Lebesgue measure) we have
[TABLE]
For completeness we give a proof of (1.2) in Appendix A.
Hence if for we define the set
[TABLE]
and define
[TABLE]
where we use to denote the Lebesgue measure of , then by (1.2) we have
[TABLE]
In fact we make:
Conjecture 1.1**.**
For each integer we have
[TABLE]
Here we are mostly interested in the structure of the set of exceptional for which (1.2) does not hold. For convenience we call the exceptional set for each and . Thus we study the exceptional sets and show that they are massive enough in a sense of Baire categories and the Hausdorff dimension.
1.3. Main results
Recall that a subset of is called *nowhere dense * if its closure in has an empty interior. We now recall the following:
Definition 1.2**.**
A subset of is of the first Baire category if it is a countable union of nowhere dense sets; otherwise it is called of the second Baire category.
For the basic properties and various applications of Baire categories we refer to [17, 19].
We now show that the complements of the sets are small.
Theorem 1.3**.**
For each and integer , the subset is of the first Baire category.
Alternatively, Theorem 1.3 is equivalent to the statement that the complement to the set
[TABLE]
is of first category. Indeed, let , . Then
[TABLE]
is a countable union of first category sets, and is of first category too. Since also for any we have , we obtain the desired equivalence.
For sets of Lebesgue measure zero, it is common to use the Hausdorff dimension to describe their size; for the properties of the Hausdorff dimension and its applications we refer to [6, 15]. We recall that for
[TABLE]
where is the Euclidean norm in .
Definition 1.4**.**
The Hausdorff dimension of a set is defined as
[TABLE]
We show that for and any the exceptional set is everywhere rich in a sense that its intersection
[TABLE]
with any cube , is of positive Hausdorff dimension, and give an explicit lower bound on this dimension.
We now define
[TABLE]
We note that
[TABLE]
and in fact if then .
Theorem 1.5**.**
For each and any cube we have
- (i)
for ,
[TABLE]
- (ii)
for ,
[TABLE]
Note that for Theorem 1.5 asserts that
[TABLE]
However Conjecture 1.1 asserts that for any and any integer we have and hence we expect
[TABLE]
and even stronger
[TABLE]
for any cube . We remark that in fact we expect for any , see Conjecture 6.2 below.
Our approach to Theorem 1.5 is based on a version of the classical Jarník–Besicovitch theorem, see [6, Theorem 10.3] or [1] and on the investigation of the distribution of large values of rational exponential sums with prime denominators. This question is of independent interest and it also gives us an opportunity to mention very interesting but perhaps not so well-known results of Knizhnerman and Sokolinskii [11, 12] about large and small values of rational exponential sums.
Furthermore, we also investigate the monomial sums
[TABLE]
to which the above technique does not apply. Similarly to the sets , for each let
[TABLE]
Similarly to Theorem 1.3 and Theorem 1.5, we also obtain the corresponding results for the monomial sums.
Theorem 1.6**.**
For each and each integer , the set is of first Baire category.
We also show the positivity of the Hausdorff dimension of
[TABLE]
for any interval . In analogy to Theorem 1.5 we have the following result.
Theorem 1.7**.**
For each and any interval , we have
- (i)
for ,
[TABLE]
- (ii)
for ,
[TABLE]
Note that for Theorem 1.7 (i), for the case , asserts that
[TABLE]
In fact only the case of is of interest as for , and this is instant from the result of Fiedler, Jurkat and Körner [7, Theorem 2].
For with Theorem 1.7 (ii), for the case , asserts that
[TABLE]
However we conjecture that for each and each one has
[TABLE]
and perhaps even stronger
[TABLE]
for any interval .
1.4. Applications to uniform distribution modulo one
A quantitative way to describe the uniformity of distribution modulo one is given by the discrepancy, see [5].
Definition 1.8**.**
Let , , be a sequence in . The discrepancy of this sequence at length is defined as
[TABLE]
Recalling that a sequence is uniform distributed modulo one if and only if the corresponding discrepancy
[TABLE]
see [5, Theorem 1.6] for a proof. We note that sometimes in the literature the scaled quantity is called the discrepancy, since our argument looks cleaner with Definition 1.8, we adopt it here.
For and the sequence
[TABLE]
we denote by the corresponding discrepancy. Motivated by the work of Wooley [23, Theorem 1.4], the authors [4] have shown that for almost all with one has
[TABLE]
In view of Lemmas 2.2 and 5.1 below, Conjecture 1.1 is equivalent to the statement that the exponent in (1.4) cannot be improved.
Thus, the bound (1.4), combined with Lemma 5.1 below, provides yet another way to obtain that
[TABLE]
holds for almost all (which is a slightly less precise version of (1.2)).
Let
[TABLE]
Theorem 1.9**.**
For each and integer the subset is of the first Baire category.
Note that this is equivalent to the statement that the complement to the set
[TABLE]
is of first Baire category.
For any cube denote .
Theorem 1.10**.**
For each and any , we have
- (i)
for ,
[TABLE]
- (ii)
for ,
[TABLE]
In the case of monomials, For we denote by the discrepancy of the sequence , and set
[TABLE]
We have the following analogues of Theorems 1.9 and 1.10
Theorem 1.11**.**
For each and integer the subset is of the first Baire category.
Furthermore, we also have the following result. For an interval denote .
Theorem 1.12**.**
For each and any interval , we have
- (i)
for ,
[TABLE]
- (ii)
for ,
[TABLE]
We remark that the case is a special case. For the linear sequence the celebrated result of Khintchine, see [5, Theorem 1.72], implies that for almost all one has
[TABLE]
2. Preliminaries
2.1. Notation and conventions
Throughout the paper, the notations , and are equivalent to for some positive constant , which throughout the paper may depend on the degree and occasionally on the small real positive parameters and .
We use to denote the cardinality of set .
The letter , with or without a subscript, always denotes a prime number.
We always identify with half-open unit cube , in particular we naturally associate Euclidean norm with points .
We say that some property holds for almost all if it holds for a set of Lebesgue measure .
We always keep the subscript in notations for our main objects of interest such as , and , but sometimes suppress it in auxiliary quantities.
2.2. Complete rational exponential sums and uniform distribution
We first recall the classical Weil bound, see, for example, [14, Chapter 6, Theorem 3]. For a prime , let denote the finite field of elements, which we identify with the set , and let . Furthermore let .
Lemma 2.1**.**
Let be a nonconstant polynomial of degree . Then we have
[TABLE]
Applying Lemma 2.1 and the completion technique (see [9, Section 12.2]) fwe derive the following bounds for incomplete sums. For one has
[TABLE]
Next, we consider discrete cubic boxes
[TABLE]
with the side length
[TABLE]
where is a set of consecutive integers, (reduced modulo if ), .
We formulate the following easy consequence of the Koksma–Szüsz inequality, see [5, Theorem 1.21].
Lemma 2.2**.**
Let , , be a sequence of vectors over and let be a box. Let
[TABLE]
Then we have
[TABLE]
where denotes the scalar product of two vectors .
2.3. Distribution of large rational exponential sums
For a vector we consider the rational exponential sum
[TABLE]
We need some results about the density of the vectors for which the sums are large.
For the answer to the question is trivial due to the following property of Gaussian sums, see [9, Equation (1.55)].
Lemma 2.3**.**
Let and with , then
[TABLE]
We now investigate the case of . For this, we define
[TABLE]
From the classical method of Mordell [16] we have
[TABLE]
Hence, taking into account the contribution from the zero vector and estimating the contribution from vectors with by Lemma 2.1, we obtain
[TABLE]
which trivially implies that
[TABLE]
Knizhnerman and Sokolinskii [11, 12] have given stronger lower bounds, asymptotically for and also for small values of , for example, .
Furthermore, by [11, Theorem 1] we have
Lemma 2.4**.**
For every integer there are some positive constants and such that
[TABLE]
for a set of cardinality .
We now show that the vectors for which the sums reach their extreme values are reasonably densely distributed. That is. we intend to show that the set of Lemma 2.4 is quite dense. Before this we provide a result on the distribution of monomial curves.
Lemma 2.5**.**
Let , . Then there exists a positive constant which depends only on such that for any box as in (2.2) with the side length we have
[TABLE]
Proof.
For a nonzero vector the Weil bound, see Lemma 2.1, gives
[TABLE]
Combining this bound with Lemma 2.2, we finish the proof. \sqcap$$\sqcup
Clearly we can replace a lower bound of Lemma 2.5 with an asymptotic formula for slightly larger values of , namely, if as . We also note that Lemma 2.5 still holds for the case .
Lemma 2.6**.**
Fix . There is an constant depending only on , such that for a box as in (2.2) with the side length , where is as in (1.3), and as in Lemma 2.4, there is .
Proof.
Adjusting if necessary, we can assume that is large enough.
Clearly, if then for any we also have . Let be an integer such that
[TABLE]
By Lemma 2.4 we conclude that there exists with for each such that
[TABLE]
For convenience we denote this set by .
Let be a box with the side length , which we decompose in a natural way as
Note that we have . Let
[TABLE]
Then Lemma 2.5 implies that
[TABLE]
provided the condition
[TABLE]
is satisfied with a sufficiently large .
We now fix a vector and consider the double exponential sums
[TABLE]
By the Cauchy-Schwarz inequality
[TABLE]
Now using that for any we have , and then changing the order of summations, we obtain
[TABLE]
By the orthogonality of exponential functions, the last sum vanishes unless for every we have . Since is a nonzero vector of , this is possible for at most pairs , and in the case the inner sum is equal to . Hence, for any nonzero vector we have
[TABLE]
Using that , we now obtain
[TABLE]
Let be the number of the vectors such that
[TABLE]
Combining the bound (2.6) with Lemma 2.2, we obtain
[TABLE]
Thus we conclude that when
[TABLE]
for some constant depending only on and . By (2.4) this condition becomes
[TABLE]
and hence it is enough to request that
[TABLE]
for a sufficiently large constant .
Combining the conditions (2.5) and (2.8), and recalling the definition of in (1.3), we conclude that there exists a large enough constant such that the inequality
[TABLE]
is sufficient to guarantee that for some we have (2.7). Since we always have when and so the result now follows. \sqcap$$\sqcup
Corollary 2.7**.**
Let be defined as in Lemma 2.4. Then for any the set
[TABLE]
is dense in .
Proof.
Let be a box of with the side length
[TABLE]
where is as in Lemma 2.6. Define
[TABLE]
By Lemma 2.5 there exists such that
[TABLE]
provided that is large enough. Thus, we conclude that
[TABLE]
Since this holds for any box of , the result follows. \sqcap$$\sqcup
Remark 2.8**.**
For the case , Corollary 2.7 follows immediately from Lemma 2.3. However in general Lemma 2.3 does not hold for and in fact with vanishing sums are often densely distributed as well.
For instance, for and a prime number with , the map: permutes . Hence, for any we have
[TABLE]
Assuming we see that
[TABLE]
By Lemma 2.5 for any box with the side length for some constant there exists such that
[TABLE]
Therefore we conclude that for any the set
[TABLE]
is a dense subset of . **
2.4. Large Weyl sums
We are going to show that the small neighbourhood of still have large exponential sums. Namely let denotes the cubic box centered at with the side length
[TABLE]
For each and a prime we define
[TABLE]
We also use from Lemma 2.4.
We use the following version of summation by parts. Let be a sequence and for each denote
[TABLE]
Let be a differential function. Then
[TABLE]
Lemma 2.9**.**
Let for some and prime . There exists an absolute constant such that if
[TABLE]
then
[TABLE]
Proof.
For any there exist such that
[TABLE]
where is the -norm in . Let . Applying summation by parts we obtain
[TABLE]
where
[TABLE]
Note that for any we have
[TABLE]
Combining with we obtain
[TABLE]
For the integral part of (2.9) we derive
[TABLE]
Thus combining with (2.10) and the definition of , and using bound (2.1) on incomplete sums, we derive
[TABLE]
where is some constant which depends on only.
Since , using the periodicity of function , we obtain
[TABLE]
Combining (2.11) and (2.12) we obtain
[TABLE]
provided
[TABLE]
for a sufficiently small constant (depending only on ), which gives the desired result. \sqcap$$\sqcup
We formulate some notation for our using on the lower bound of the Hausdorff dimension of .
Lemma 2.10**.**
Let . For any there exists such that for any and any cubic box with the side length there exists a box with the side length and such that for , where is as in Lemma 2.9, and all , we have
[TABLE]
Proof.
Let be the box. For the box , Lemma 2.6 implies that there exists a point
[TABLE]
provided is large enough. Let The condition gives , and hence .
By the choice of and the condition , Lemma 2.9 implies that for all we have
[TABLE]
which gives the desired result. \sqcap$$\sqcup
Definition 2.11** (-patterns).**
Let and . Let be a box with with the side length . We divide the box into smaller boxes in a natural way. For each of these boxes we pick a smaller box, at an arbitrary location with the side length . The resulting configuration of boxes with the side length is called an -pattern.**
An illustrative example of an -pattern is given in Figure 2.1.
We note that each -pattern is a subset of . For our applications we find -patterns such that the Weyl sums are large inside of the small boxes. We show that for any box there are -patterns which admit large Weyl sums. More precisely we have the following.
Lemma 2.12**.**
Let and be the same as in Lemma 2.10. Let and with the side length . There exists such that and . Furthermore there exists a -pattern, which we denote by , such that for
[TABLE]
and all we have
[TABLE]
Proof.
Since , we divide the box into smaller boxes of equal sizes in a natural way. We label them by for convenience.
For each , , Lemma 2.10 asserts that there exists a box with the side length , and for all we have the desired bound.
We finish the proof by taking . \sqcap$$\sqcup
2.5. Hausdorff dimension of a class of Cantor sets
By a repeated application of Lemma 2.12, we find large Weyl sums on a Cantor-like set. This implies a lower bound for the Hausdorff dimension of . In this section we investigate a general construction of Cantor-like sets.
Now we show the construction of the Cantor sets by iterating the construction of -patterns.
Let
[TABLE]
such that for each , we have
[TABLE]
For convenience we also denote
[TABLE]
the side length of .
For each we ask that the triple satisfies the condition on in Definition 2.11. In particular, we always assume that
[TABLE]
and we denote
[TABLE]
for every .
We start from the cube and take a -pattern inside of .
Let be the collection of these boxes. More precisely let
[TABLE]
For each we take a -pattern inside of , and we denote these sub-boxes of by with . Let
[TABLE]
Figure 2.2 shows an example of this construction.
Suppose now we have which is a collection of boxes with the side length . For each of these box we take a -pattern inside of the box . Let be the collections of these boxes, that is
[TABLE]
Our Cantor-like set is defined by
[TABLE]
where
[TABLE]
There are many possible outcomes by the above construction, we let denote all possible patterns.
From our construction clearly we have , and is a compact set, and hence is a nonempty compact set. Furthermore we obtain the lower bound of these Cantor sets by using the following mass distribution principle [6, Theorem 4.2].
Lemma 2.13**.**
Let and let be a measure on such that . If for any box with for some we have
[TABLE]
then the Hausdorff dimension of is at least .
Lemma 2.14**.**
Let and let , , are given by (2.14). Then
[TABLE]
Proof.
We show the the upper bound of first. Let
[TABLE]
Then there exists a sequence , , such that
[TABLE]
The construction of implies for each
[TABLE]
Thus for any we obtain
[TABLE]
The definition of Hausdorff dimension, see Definition 1.4, implies that . By the arbitrary choices of and we obtain the upper bound
[TABLE]
Now we turn to the lower bound of . We first define a measure on (natural measure). For each and any subset let
[TABLE]
where is the indicator function of a set . Observe that for each we have
[TABLE]
We note that the measure weakly convergence to a measure , see [15, Chapter 1].
Let then there exists such that for any we have
[TABLE]
Let with . Then there exists such that
[TABLE]
Observe that
[TABLE]
Applying , we obtain
[TABLE]
Combining with the estimate (2.15) and the condition , we have
[TABLE]
Applying the mass distribution principle given in Lemma 2.13, we have . By the arbitrary choice of we obtain that , which finishes the proof. \sqcap$$\sqcup
2.6. Monomial exponential sums
We need the following elementary statement, see, for example [13, Equation (82)] for a more general statement.
Lemma 2.15**.**
Let with , then for integer
[TABLE]
One can certainly adapt the arguments in the proof of Lemma 2.9 to get a lower bound on . However we can achieve better results with the following approximate formula of Vaughan [20, Theorem 4.1].
Lemma 2.16**.**
Let
[TABLE]
with some relatively prime integers and . Then
[TABLE]
We now easily see that Lemma 2.16 implies the following result.
Lemma 2.17**.**
Let and let be a prime number such that . Let with for some . There exists an absolute constant such that for any If
[TABLE]
then
[TABLE]
provided that is large enough.
Proof.
Using Lemma 2.16 with we see that the assumed upper bound on implies that
[TABLE]
Hence taking small enough we obtain
[TABLE]
Therefore by Lemmas 2.15 and 2.16
[TABLE]
Recalling the lower bound we see that the first term dominates, which finishes the proof. \sqcap$$\sqcup
3. Proofs of abundance of large Weyl sums
3.1. Proof of Theorem 1.3
The idea is that we first show that the exponential sums are large at a dense subset of , and then we show the exponential sums are still large at the small neighbourhoods of these points. This implies that the subset has large topology for each .
Let the sets be as in Lemma 2.9.
For positive integers and we consider the sets
[TABLE]
and define
[TABLE]
Using Lemma 2.9, with , we conclude that for each we have
[TABLE]
Let and be an arbitrary open cubic box. Then Corollary 2.7 implies that there exists an open cubic box such that . It follows that is a nowhere dense subset. Furthermore since
[TABLE]
we obtain that the set is the countable union of nowhere dense sets, and hence is of first category. Together with (3.1) we complete the proof.
3.2. Proof of Theorem 1.5
3.2.1. Preamble
We first note that our methods for the cases and are different. For the case we use Lemma 2.3. As it is shown in Remark 2.8, in general Lemma 2.3 does not hold for , for this case we use the results from Section 2.4.
Throughout the proof we fix the cube . In particular, all implied constants may depend on .
We use to denote the distance in the -norm between and the closest point .
3.2.2. Case (i): .
For we define
[TABLE]
The classical Jarník–Besicovitch theorem, see [6, Theorem 10.3] or [1], asserts that
[TABLE]
We note that the method in the proof of [6, Theorem 10.3] (or see the proof of Lemma 3.1) imply that
[TABLE]
For our purpose we need obtain an analogy of (3.2) for .
We introduce some notation first. For a prime number we define
[TABLE]
where is the -norm in , and
[TABLE]
Applying the arguments of [6, Theorem 10.3] to our setting we have the following.
Lemma 3.1**.**
Using the above notation for any we have
[TABLE]
Proof.
For the upper bound first note that for each the set can be covered by at most boxes with the side length . Since for each
[TABLE]
and for any we have
[TABLE]
Definition 1.4 implies . By the arbitrary choice of we conclude
[TABLE]
Now we turn to the lower bound. Let be a sequence rapidly increasing prime numbers such that
[TABLE]
For each define
[TABLE]
An important fact is that for different primes the sets and are disjoint when is large enough. Indeed, this follows from the choice of and that for and ,
[TABLE]
Note that there are prime numbers between and , and for each prime number the set contains boxes with the side length , which due to the fact that the cube is fixed. Thus the set consists of boxes with the side length . We remark that the implied constant may depend on , however it is not hard to see that for a fixed cube this constant does not affect the result. Let
[TABLE]
We claim that
[TABLE]
We show some explanation in the following. For each let
[TABLE]
Note that . An important fact is that for any box of with the side length it contains
[TABLE]
uniformly distributed boxes of with the side length . Denote . It follows, also using (3.4), that contains at least
[TABLE]
boxes with the side length .
By giving a measure on in a similar way as in the proof of Lemma 2.14, and then applying the mass distribution principle, see Lemma 2.13, we obtain
[TABLE]
which proves the claim (3.5).
Observe that for each there are infinitely such that , and hence and . By the monotonicity property of the Hausdorff dimension we see from (3.5) that
[TABLE]
which together with (3.3) finishes the proof. \sqcap$$\sqcup
To conclude the proof for the case , it is sufficient to prove with some , since
[TABLE]
Let then there exists with such that
[TABLE]
Applying Lemma 2.3, exactly as in the proof of Lemma 2.9 we see that
[TABLE]
provided
[TABLE]
for some absolute constant .
Furthermore, for any small , if we have
[TABLE]
then we also have
[TABLE]
Note that the implied constant here does not depend on . Clearly we can find satisfying (3.7) and (3.8) simultaneously provided that
[TABLE]
and is large enough. It follows that for each with large enough there exists such that
[TABLE]
This implies that . Combining with (3.6) and (3.9) we obtain that
[TABLE]
By the arbitrary choice of small and positive , we finish the proof.
3.2.3. Case (ii): .
We note that our method also works for , thus we only assume in the following.
Let be a sequence rapidly increase prime numbers such that
[TABLE]
Let such that
[TABLE]
As before, we define as the side length of , that is, as in (2.13). For each let
[TABLE]
and choosing large enough, we see that we can assume that .
Fix some sufficiently small and for each let
[TABLE]
where is given by (1.3), such that . For example, the choice
[TABLE]
is satisfactory since we may choose such that for any small .
Denote
[TABLE]
Applying Lemma 2.14 to the sequences we obtain the following.
Lemma 3.2**.**
In the above notation (3.12) and (3.14) and under the conditions (3.10), (3.11) and (3.13), for any , we have
[TABLE]
Proof.
Recalling (3.10) and (3.13), we obtain
[TABLE]
and
[TABLE]
Lemma 2.14 gives
[TABLE]
which finishes the proof. \sqcap$$\sqcup
We are now going to show that there exists a pattern such that for some which may depend on and . Thus Lemma 3.2 implies that
[TABLE]
Our construction is inductive.
For given by (2.13) and with
[TABLE]
(note that we request ), by Lemma 2.12 there exists a -pattern, which we denote by , such that for
[TABLE]
and all we have
[TABLE]
Now, suppose that we have a pattern which is a collection of boxes with the side length . For each box again by Lemma 2.12 there exists a -pattern such that for
[TABLE]
and all we have
[TABLE]
Let
[TABLE]
For convenience we use the same notation to denote
[TABLE]
Let
[TABLE]
Then by (3.16) we conclude that
[TABLE]
provided that
[TABLE]
and the condition (3.11) holds.
The inequalities (3.11) and (3.17) imply that it is sufficient to take any such that
[TABLE]
Combining this with (3.15), and using that we obtain
[TABLE]
Since this lower bound holds for any , we conclude the proof of Theorem 1.5.
4. Proofs of abundance of large monomial sums
4.1. Proof of Theorem 1.6
For and some we define the sets
[TABLE]
and
[TABLE]
Let . Applying Lemma 2.17 we see that
[TABLE]
provided that
[TABLE]
for some and sufficiently large , where is an absolute constant.
Furthermore, for each if we have
[TABLE]
then we also have
[TABLE]
By conditions (4.3) and (4.4) we conclude that for any such that
[TABLE]
there exists such that the conditions (4.3) and (4.4) hold simultaneously.
It follows that there exists some such that for any
[TABLE]
Therefore if (4.5) holds then
[TABLE]
For each let
[TABLE]
Clearly for each the set is an open and dense subset of , and hence is a nowhere dense subset of . Therefore we obtain that the set
[TABLE]
is of first Baire category set. Now from (4.2) and (4.6) we obtain
[TABLE]
and hence we finish the proof.
4.2. Proof of Theorem 1.7
4.2.1. Preamble
We note that for the monomials the methods for the cases and are also different. For the case we use Lemma 2.3, while for the case we use Lemma 2.15.
Throughout the proof we fix the interval . In particular, all implied constants may depend on .
4.2.2. Case (i): .
This case follows by applying the similar arguments to the proof of Theorem 1.5 for the case .
For and some let
[TABLE]
and
[TABLE]
As we claimed before that the method in the proof of [6, Theorem 10.3] (or see the proof of Lemma 3.1) imply that
[TABLE]
Applying Lemma 2.3 and Lemma 2.9 we conclude that for any there exists such that
[TABLE]
provided that
[TABLE]
Note that this is the same condition as (3.9) up to the small parameter . Under this condition for the parameter we conclude . Combining with (4.7) we obtain the desired result.
4.2.3. Case (ii): .
We slightly modify the definition of the set in (4.1) by using instead of , that is, we now set
[TABLE]
while the set is still defined by (4.2).
By adapting the arguments of [6, Theorem 10.3] and Lemma 3.1 to the sets we have the following.
Lemma 4.1**.**
Using the above notation for any we have
[TABLE]
Proof.
Let . Note that for any we have
[TABLE]
Since
[TABLE]
Definition 1.4 implies . By the arbitrary choice of we conclude that
[TABLE]
Now we turn to the lower bound of . Let be a sequence rapidly increasing prime numbers satsifying (3.4). For each let
[TABLE]
and
[TABLE]
Clearly we have
[TABLE]
Hence, it is sufficient to show that
[TABLE]
Let be two distinct prime numbers with , and let and such that . Then
[TABLE]
and
[TABLE]
Since , we conclude that the sets and are disjoint for two distinct prime numbers when is large enough.
Note that there are prime numbers between and , and for each prime number the set contains intervals with length (since the interval is fixed). Thus the set consists of intervals with length nearly . As in the proof of Theorem 1.5, we remark that the implied constant may depend on , however it is not hard to see that for a fixed interval this constant does not affect the result.
By (3.4), each interval of consists nearly intervals of of length .
Applying the method in [6, Example 4.7], see also Lemma 3.1, we obtain the inequality (4.10) which together with (4.8) and (4.9) concludes the proof. \sqcap$$\sqcup
For each we intend to find some such that
[TABLE]
Hence, by the monotonicity property of the Hausdorff dimension and Lemma 4.1 we obtain
[TABLE]
Applying the arguments in the proof of Theorem 1.6, see (4.5), we obtain that for any
[TABLE]
and any there exists some such that for any
[TABLE]
Thus the condition of Lemma 4.1 is satisfied. Combining with (4.11), we obtain
[TABLE]
which finishes the proof.
5. Proofs of abundance of poorly distributed polynomials
5.1. Exponential sums and the discrepancy
For our applications we need the following Koksma-Hlawlka inequality, see [5, Theorem 1.14] for a general statement.
Lemma 5.1**.**
Using the above notation, for any
[TABLE]
Note that in particular, Lemma 5.1 implies for .
5.2. Proof of Theorems 1.9 and 1.11
We see that Lemma 5.1 implies that for any cube or interval and any one has
[TABLE]
Combining (5.1) for and with Theorems 1.3 and 1.6 we obtain Theorems 1.9 and 1.11, respectively.
5.3. Proof of Theorems 1.10 and 1.12
Applying (5.1) and the monotonicity property of Hausdorff dimension we have
[TABLE]
and
[TABLE]
Combining this with Theorems 1.5 and 1.7 we obtain Theorems 1.10 and 1.12, respectively.
6. Further results, open problems and conjectures
6.1. Further extensions of Theorems 1.3 and 1.5
On the other hand, the method of proof of Lemma 2.6 is quite robust and can be implies to some other families of polynomials, such as sparse polynomials
[TABLE]
In turn, this can be used to obtain versions of Theorems 1.3 and 1.5 for exponential sum with sparse polynomials
[TABLE]
where with . More precisely, for each and , we define
[TABLE]
We note that (2.3) can easily be extended to sparse polynomials
[TABLE]
which in turn leads to full analogues of Lemmas 2.4, 2.5 and 2.6. Then we have the following direct generalisations of Theorems 1.3 and 1.5 which can be obtained at the cost of essentially only typographical changes in their proofs. For each and with ,
- (A)
the subset is of the first Baire category;
- (B)
for any cube we have,
[TABLE]
where is given by (1.3). Note that we recover the bound of Theorem 1.5 (for ) provided .
Remark 6.1**.**
We note that (6.1) is only a lower bound rather than an asymptotic formula as (2.3). In fact, most likely an asymptotic form of (6.1) holds with instead of , see [21]. However this is inconsequential for our results.
We note that it is natural to try to improve Theorem 1.5 via an appropriate version of Lemma 2.16 for arbitrary polynomials. Unfortunately the only known result in this direction [20, Theorem 7.2] is not strong enough to lead to such an improvement.
6.2. Further questions about the structure of Weyl sums
For we now define
[TABLE]
Alternatively, we may also define
[TABLE]
By the definition we have
[TABLE]
For each we define the level set
[TABLE]
Clearly these sets form a decomposition of . There are several natural questions about these sets. Note that Conjecture 1.1 asserts that for any we have . We may make the following stronger conjecture.
Conjecture 6.2**.**
For we have
[TABLE]
We may also use the Hausdorff dimension to measure the size of .
Question 6.3**.**
What is the Hausdorff dimension of ?
Finally, one can also ask whether the function which is defined by (6.2) has multifractal structure. More precisely we ask the following:
Question 6.4**.**
Does there exist a set with such that for any we have
[TABLE]
6.3. Further questions about the distribution of large complete rational sums and possible improvements of Theorem 1.5
It is certainly natural to consider more general transformations
[TABLE]
instead of just which is essentially used in the proof of Lemma 2.6. The transformation (6.3) is very similar to the transformation used in the proof of [18, Lemma 4]. However, while in [18] the Deligne bound (see [9, Section 11.11]) is applied to the corresponding double exponential sums with a polynomials in and , in the case of (6.3) these polynomials are singular, and so the Deligne bound does not apply. It is certainly interesting to find an alternative way, and thus improve Lemma 2.6, in which can possibly be replaced with .
Lemma 2.5 study the distribution of sets
[TABLE]
where for each . Lemma 2.5 asserts that for any box of with the side length for some large constant there exists such that
[TABLE]
Note that there are totally vectors
[TABLE]
thus the smallest in Lemma 2.5 should be
[TABLE]
One could ask that is this a sufficient condition.
Question 6.5**.**
Let . Is it true that for any there exists a constant such that any box of with the side length contains a vector for some ?
It is also interesting to consider the special case that is the distribution of
[TABLE]
Note that studying the distribution of
[TABLE]
is already an interesting and hard problem related to the distribution of quadratic nonresidues.
A possible approach to improving Theorem 1.5 is via finding an asymptotic formula or at least a lower bound for the average of over small box as in (2.2). In fact finding lower bounds for the moments
[TABLE]
of nontrivial sums with is of independent interest. For one can easily extend the result of Mordell [16], that is, (2.3), to any and obtain
[TABLE]
where
[TABLE]
see also [12, Equation (2)].
Using the same arguments as in the proof of Lemmas 2.5 and 2.6 with , one can obtain an asymptotic formula
[TABLE]
see Appendix B, which is nontrivial in the case of cubes with the side length for any fixed . However we are interested in much smaller boxes, for example of size of the side length about . In fact, a lower bound of the form for any fixed is sufficient for our applications.
6.4. An approach to Conjecture 1.1
Recall that Conjecture 1.1 asserts that for each integer , and the bound (1.2) gives . Thus it is sufficient to prove that for any one has .
For and integer we define
[TABLE]
We can write
[TABLE]
Lemma 6.6**.**
Let then , and hence
[TABLE]
Proof.
Applying the trivial bound we obtain
[TABLE]
Combining with Parseval identity
[TABLE]
and the condition , we obtain the desired result. \sqcap$$\sqcup
Suppose that the sets are pair independent with respect to the Lebesgue measure , i.e., for any we have
[TABLE]
then the Borel-Cantelli lemma and (6.6) implies that . Surely the pair independent assumption is not true, and an ordinary way to overcome this is by the following arguments. One first show that these sets are weak independent, that is there exists some constant such that for any we have
[TABLE]
then a variant of the Borel-Cantelli lemma gives
[TABLE]
Secondly one may use a zero-one law to pass from to .
Appendix A Proof of the bound (1.2) and some extensions
By applying a very special case of the Menshov–Rademacher theorem, see [10, p. 251] for the general statement, we conclude that if for some sequence of complex numbers we have
[TABLE]
then the series
[TABLE]
converges for almost all .
For we have
[TABLE]
It follows that for any the series
[TABLE]
converges for almost all . Together with the Fubini theorem, we obtain that the series
[TABLE]
converges for almost all .
Now we turn to the proof of (1.2). We denote
[TABLE]
and
[TABLE]
Fix any , and write
[TABLE]
Then the summation by parts gives
[TABLE]
where
[TABLE]
Since the condition (A.1) is satisfied, for almost all there exits some positive such that for all we have
[TABLE]
Substituting (A.3) in (A.2) we easily conclude that for almost all we have (1.2).
We note that the above arguments implies that for any the bound
[TABLE]
holds for almost all .
Furthermore, one can easily see that the above argument work in a much broader generality. For example, let be functions such that for any we have for each . If one of these functions is eventually strictly monotonic, then for almost all we have
[TABLE]
For instance, for the bound
[TABLE]
holds for almost all .
Remark A.1**.**
For the case we can obtain the bound for the estimate (1.2) in a different way. The Khinchine theorem, see [1, Introduction], implies that for almost all irrational there exits some positive constant such that for all rational with we have
[TABLE]
On the other hand, by [9, Theorem 8.1], if with and then for any one has
[TABLE]
Combining these two results, we conclude that for almost all one has
[TABLE]
Appendix B Moments of rational exponential sums over small boxes
Here we sketch a proof of (6.5). Clearly we can assume that (it is easy see that by Lemma 2.1 discarding such sums changes the value of by , which can be absorbed in the error in (6.5)). In particular, we can assume that .
Observe that for any and we have
[TABLE]
where
[TABLE]
It follows that
[TABLE]
where
[TABLE]
Let be the set of with where is as in (2.2). Hence for , we have
[TABLE]
By the orthogonality of characters, and then changing the order of summation and separating the contribution from we obtain
[TABLE]
where
[TABLE]
We note that if the first coordinate of is zero. Combining (B.1) and (B.2), we obtain
[TABLE]
where
[TABLE]
By Lemma 2.1 we obtain
[TABLE]
Hence, recalling (6.4) we obtain
[TABLE]
To estimate we note that by [9, Equation (8.6)] we have
[TABLE]
By Lemma 2.1 we now see that
[TABLE]
Using the Cauchy inequality, as in the proof of Lemma 2.6, we have
[TABLE]
Hence,
[TABLE]
which together with (B.3) yields (6.5).
Acknowledgement
The authors are grateful to Fernando Chamizo, Boris Kashin, Bryce Kerr, Sergei Konyagin and Trevor Wooley for helpful advice and discussions.
This work was supported in part by ARC Grant DP170100786.
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