On polynomial configurations in fractal sets
We show that subsets of of large enough Hausdorff and Fourier dimension contain polynomial patterns of the form
where are real matrices, is a real polynomial in variables and .
In this work we investigate the presence of point configurations in subsets of which are large in a certain sense. When is a subset of of positive Lebesgue measure, a consequence of the Lebesgue density theorem is that contains a similar copy of any finite set. A more difficult result of Bourgain  states that sets of positive upper density in contain, up to isometry, all large enough dilates of the set of vertices of any fixed non-degenerate -dimensional simplex. In a different setting, Roth’s theorem  in additive combinatorics states that subsets of of positive upper density contain non-trivial three-term arithmetic progressions.
When a subset is only supposed to have a positive Hausdorff dimension, a direct analogue of Roth’s theorem is impossible. Indeed Keleti  has constructed a set of full dimension in not containing the vertices of any non-degenerate parallelogram, and in particular not containing any non-trivial three-term arithmetic progression. Maga  has since extended this construction to dimensions . The work of Łaba and Pramanik  and its multidimensional extension by Chan et al.  circumvent these obstructions under additional assumptions on the set , which we now describe.
When is a compact subset of , Frostman’s lemma [46, Chapter 8] essentially states that its Hausdorff dimension is equal to
where is the space of probability measures supported on . On the other hand, the Fourier dimension of is
It is well-known that we have for every compact set , with strict inequality in many instances, and we call a Salem set when equality holds. There are various known constructions of Salem sets [39, 25, 7, 8, 24, 28, 22], several of which [27, 11] also produce sets with prescribed Hausdorff and Fourier dimensions .
In a very abstract setting, one may ask whether it is possible to find translation-invariant patterns of the form
in the product set , where the are certain shift functions. When , the map considered is often a submersion of an open subset of onto , and then one can find a pattern of the desired kind in via the implicit function theorem. A natural restriction is therefore to assume that in this multidimensional setting. Chan et al.  studied the case where the maps are linear for matrices , generalizing the study of Łaba and Pramanik for three-term arithmetic progressions, under the following technical assumption.
Let and suppose that with and . We say that the system of matrices is non-degenerate when
for every set of indices , with the convention that .
Requirements similar to the above arise when analysing linear patterns by ordinary Fourier analysis in additive combinatorics , and there is a close link with the modern definition of linear systems of complexity one . The main result of Chan et al.  gives a fractal analogue of the multidimensional Szemerédi theorem  for non-degenerate linear systems, when the Frostman measure has both dimensional and Fourier decay. We only state it in the case where divides for simplicity.
Theorem 1.2 (Chan, Łaba and Pramanik).
Let , and . Suppose that is a compact subset of and is a probability measure supported on such that111 In fact, this theorem was proved in  under the more restrictive condition for a fixed constant . However, by examining the proof there, one can see that the constant may be allowed to grow polynomially in , as was the case in the original argument of Łaba and Pramanik .
for all , and . Suppose that is a non-degenerate system of matrices in the sense of Definition 1.1. Assume finally that with and, for some ,
for a sufficiently small constant . Then, for every collection of strict subspaces of , there exists such that
Note that the Hausdorff dimension is required to be large enough with respect to the constants involved in the dimensional and Fourier decay bounds for the Frostman measure. A construction due to Shmerkin  shows that the dependence of on the constants cannot be removed.
In practice, Salem set constructions provide a family of fractal sets indexed by , and it is often possible to verify the conditions of Theorem 1.2 for close to ; this was done in a number of cases in . The requirement of Fourier decay of the measure serves as an analogue of the notion of pseudorandomness in additive combinatorics , under which we expect a set to contain the same density of patterns as a random set of same size.
In this work we consider a class of polynomial patterns, which generalizes that of Theorem 1.2. We aim to obtain results similar in spirit to the Furstenberg-Sarközy theorem [40, 14] in additive combinatorics, which finds patterns of the form in dense subsets of . A deep generalization of this result is the multidimensional polynomial Szemerédi theorem in ergodic theory [4, 6] (see also [5, Section 6.3]), which handles patterns of the form (1.1) where each shift function is an integer polynomial vector with zero constant term. By contrast, the class of patterns we study includes only one polynomial term, which should satisfy certain non-degeneracy conditions. We are also forced to work with a dimension , and all these limitations are due to the inherent difficulty in analyzing polynomial patterns through Fourier analysis. On the other hand, we are able to relax the Fourier decay condition on the fractal measure needed in Theorem 1.2.
Let , and . Suppose that is a compact subset of and is a probability measure supported on such that
for all , and . Suppose that is a non-degenerate system of real matrices in the sense of Definition 1.1. Let be a real polynomial in variables such that and the Hessian of does not vanish at zero. Assume furthermore that, for a constant ,
for a sufficiently small constant . Then, for every collection of strict subspaces of , there exists such that
Our argument follows broadly the transference strategy devised by Łaba and Pramanik , and its extension by Chan and these two authors . However, the case of polynomial configurations requires a more delicate treatment of the singular integrals arising in the analysis. The weaker condition on is obtained by exploiting restriction estimates for fractal measures due to Mitsis  and Mockenhaupt . A more detailed outline of our strategy can be found in Section 3. By the method of this paper, one can also obtain an analogue of Theorem 1.2 with the same relaxed condition on the exponent , and we state this version precisely in Section 9.
For concreteness’ sake, we highlight the lowest dimensional situation handled by Theorem 1.3. When and , this theorem allows us to detect patterns of the form
for matrices of full rank such that has full rank, and for a non-degenerate quadratic form in three variables. We may additionally impose that by setting in Theorem 1.3. For example, when , and , we can detect the configuration
with . However, we cannot detect the configuration
for then we have and , and the condition is not satisfied.
Note also that, in the statement of Theorem 1.3, one may add a linear term in variables to the polynomial without affecting the assumptions on it. This allows for some flexibility in satisfying the matrix non-degeneracy conditions of Definition 1.1, since one may alter the last line of at will. For example, the degenerate system of matrices , and the polynomial give rise to the configuration
Rewriting , we see that still has non-degenerate Hessian at zero and the configuration is now associated to the system of matrices , , which is easily seen to be non-degenerate. One possible explanation for this curious phenomenon is that, by comparison with the setting of Theorem 1.2, we have an extra variable at our disposition, since .
Finally, we note that there is a large body of literature on configurations in fractal sets where Fourier decay assumptions are not required. Here, the focus is often on finding a large variety (in a specified quantitative sense) of certain types of configurations. A well-known conjecture of Falconer [46, Chapter 9] states that when a compact subset of has Hausdorff dimension at least , its set of distances must have positive Lebesgue measure. This can be phrased in terms of containing configurations with for all , where is “large.” Wolff  and Erdog̃an [13, 12] proved that the distance set has positive Lebesgue measure for , and Mattila and Sjölin  showed that it contains an open interval for . More recently, Orponen  proved using very different methods that has upper box dimension 1 if is -Ahlfors-David regular with . There is a rich literature generalizing these results to other classes of configurations, such as triangles , simplices [17, 20], or sequences of vectors with prescribed consecutive lengths [3, 21].
In a sense, the configurations studied in these references enjoy a greater degree of directional freedom, which ensures that they are not avoided by sets of full Hausdorff dimension. By contrast, a Fourier decay assumption is necessary to locate -term progressions in a fractal set of full Hausdoff dimension (as mentioned earlier), and in light of recent work of Mathé , it is likely that a similar assumption is needed to find polynomial patterns of the form (1.2). It is, however, possible that our non-degeneracy assumptions are not optimal, or that special cases of our results could be proved without Fourier decay assumptions222 After this article was first submitted for publication, a result of this type was indeed proved by Iosevich and Liu .. Loosely speaking, we would expect that configurations with more degrees of freedom are less likely to require Fourier conditions, but the specifics are far from understood and we do not feel that we have sufficient data to attempt to make a conjecture in this direction.
Acknowledgements. This work was supported by NSERC Discovery grants 22R80520 and 22R82900.
We define the following standard spaces of complex-valued functions and measures:
Similar notation is employed for functions on . We write for . We let denote either the Lebesgue measure on or the normalized Haar measure on . We let denote generically the Euclidean surface measure on a submanifold of . When is a function on an abelian group and is an element of , we denote the -shift of by . When is a matrix we denote its transpose by . We also write for an integer and .
3. Broad scheme
In this section we introduce the basic objects that we will work with in this paper. We also state the intermediate propositions corresponding to the main steps of our argument, and we derive Theorem 1.3 from them at the outset.
We fix a compact set and a probability measure supported on . For technical reasons, we suppose that . We fix two exponents , as well as two constants , where the subscript in the second constant indicates that it is allowed to vary with . We assume that the measure verifies the following dimensional and Fourier decay conditions:
We suppose that the second constant involved blows up (if at all) at most polynomially as tends to :
We also let and we consider smooth functions , where is an open neighborhood of zero. We are interested in locating the pattern
in . While this abstract notation is sometimes useful, in practice we work with the maps
where is a non-degenerate system of matrices in the sense of Definition 1.1 and is such that and the Hessian of does not vanish at zero. We also fix a smooth cutoff supported on such that on a small box and the Hessian of is bounded away from zero on the support of . This cutoff is used in Definition 3.2 below. We take the opportunity here to state an equivalent form of Definition 1.1 when .
If , we say that the system of matrices with is non-degenerate when, for every and writing , the matrices
(where the hat indicates omission) have rank .
We also state a few notational conventions applied throughout the article. When is a system of matrices, we define the matrix by . Unless mentioned otherwise, we allow every implicit or explicit constant in the article to depend on the integers , the constant , the matrices and the polynomial , and the cutoff function . This convention is already in effect in the propositions stated later in this section.
We start by defining a multilinear form which plays a central role in our argument.
Definition 3.2 (Configuration form).
For functions , we let
In Section 4, we show that the multilinear form has the following convenient Fourier expression.
For measurable functions on and on , we let
whenever the integral is absolutely convergent or the integrand is non-negative. For every , we have
where is the oscillatory integral of Definition 4.1.
We may extend the configuration operator to measures, whenever we have absolute convergence of the dual form.
Definition 3.4 (Configuration form for measures).
The next step, carried out in Section 5, is to obtain bounds for the dual multilinear form evaluated at the Fourier-Stieljes transform of the fractal measure . Such bounds hold only in certain ranges of and under certain restrictions on .
Let and suppose that for a constant small enough with respect to ,
Recalling Definition 3.4, we see that is well-defined under the conditions (3.7). In practice, we will need slight variants of Proposition 3.5, which are discussed in Section 5. In the same section, we obtain singular integral bounds for bounded functions of compact support.
Suppose that . Then there exists depending at most on such that the following holds. For functions with support in ,
Let and suppose that (3.7) holds. Then there exists a measure such that
is supported on the set of such that:
for every hyperplane .
In Section 7, we show how to obtain a positive mass of polynomial configurations in sets of positive density, through the singular integral bound of Proposition 3.6 and the arithmetic regularity lemma from additive combinatorics.
Suppose that . Then, uniformly for every function such that , and , we have
In Section 8, we show how to obtain a positive mass of configurations by a transference argument, by which the fractal measure is replaced by a mollified version of itself which is absolutely continuous with bounded density, allowing us to invoke Proposition 3.8.
Let and suppose that
for a sufficiently small constant . Then
At this stage we have stated all the necessary ingredients to prove the main theorem.
Proof of Theorem 1.3. We may assume that after a translation and dilation, which does not affect the assumptions on except for the introduction of constant factors in bounds. By Proposition 3.7 , there exists a measure with mass supported on
and such that for every collection of hyperplanes of . We have therefore proven the result if we can show that , for then and the set cannot be empty. We may apply Proposition 3.9 to obtain precisely this conclusion when is close enough to with respect to (and the other implicit parameters ). ∎
To conclude this outline, we comment briefly on the role that the Fourier decay hypothesis plays in our argument. Using the restriction theory of fractals, the assumption (3.2) is used together with the ball condition (3.1) in Appendix B to deduce that for an arbitrary , provided that is close enough to (depending on ). The Hausdorff dimension condition (3.1) alone does yield information on the average Fourier decay of , via the energy formula [46, Chapter 8], but this type of estimate seems to be insufficient to establish the boundedness of the singular integrals we encounter. Section 5 on singular integral bounds and Section 7 on absolutely continuous estimates only use the Fourier moment bound above. On the other hand, the estimation of degenerate configurations in Section 6 and the transference argument of Section 8 exploit in an essential way the assumption of uniform Fourier decay.
4. Counting operators and Fourier expressions
In this section we describe the various types of pattern-counting operators and singular integrals that arise in trying to detect translation-invariant patterns in the fractal set of the introduction. First, we define an oscillatory integral which arises naturally in the Fourier expression of the configuration form in Definition 3.2.
Definition 4.1 (Oscillatory integral).
For and we define
We now derive the dual expression of the configuration form announced in Section 3.
Proof of Proposition 3.3. By inserting the Fourier expansions of and by Fubini, we have
We single out a useful bound for the configuration operator, typically used when the are either the measure or a mollified version of it.
For measures , we have
where the left-hand side is absolutely convergent if the right-hand side is finite.
This follows from Definition 3.4 and the successive bounds
In some instances we will need a slightly more general multilinear form, as follows.
Definition 4.3 (Smoothed configuration form).
For functions and , let
For functions and , we have
By inserting the Fourier expansions of and by Fubini, we obtain
5. Bounding the singular integral
This section is devoted to the central task of bounding the singular integral (3.6), when the kernel involved is the oscillatory integral from Definition 4.1. We will rely crucially on the following decay estimate.
Assuming that the neighborhood of zero has been chosen small enough, we have
By Definition 4.1, we have , where
Consider the hypersurface of , then our assumptions on mean that has non-zero Gaussian curvature. Observe that is the Fourier transform of , where is the surface measure on and is a smooth function with same support as . Therefore it satisfies the decay estimate [42, Chapter VIII]
uniformly in , which concludes the proof. ∎
The main result of this section is a bound on the singular integral for functions in , for a range of depending on . In practice we will apply the proposition below when is close to , which requires the parameter to be larger than , and when the functions are powers of or bounded functions supported on .
Let and , and write
Let be non-negative measurable functions on . Provided that
we have, uniformly in ,
The first step towards the proof of this proposition is to bound moments of the kernels on certain subspaces. Consider the linear maps given by
For every and , the set is an affine subspace of of dimension . Recall that , so that in the regime we expect to have bounded moments of order on each of the subspaces , under reasonable non-degeneracy conditions on the matrix . As the next lemma shows, what is needed is precisely the content of Definition 3.1.
Let and suppose that . Then for we have, uniformly in and ,
First note that the assumptions of Definition 3.1 mean that is injective on for . To see that, observe that the conditions
can be put in matrix form
Since , the matrices above have empty kernel if and only if they have rank , a set of conditions which is easily seen to be equivalent to that of Definition 3.1.
We parametrize the affine subspace by , where runs over , is picked such that , and is a rotation mapping the subspace to . We obtain