Let be a sequence of independent, identically distributed random variables with mean . The strong law of large numbers asserts that

almost surely. Without loss of generality, one can assume that are mean-zero by defining . If we further assume a finite variance, that is , the Hartman-Wintner  law of the iterated logarithm gives an exact error estimate for the strong law of large numbers. More precisely,

where the constant can not be replaced by a smaller constant. That is, the quantity gets as large/small as infinitely often. The purpose of our current work is to prove a more delicate variational asymptotic that refines the law of the iterated logarithm and captures more subtle information about the oscillations of a sums of i.i.d random variables about its expected value. More precisely,

Theorem Let be a sequence of independent, identically distributed mean zero random variables with variance and satisfying .  If we let denote the set of all possible partitions of the interval into subintervals, then we have almost surely:

 .

Choosing the partition , to contain a single interval  immediately recovers  the upper bound in the law of the iterated logarithm. This result also strengthens earlier work of J. Qian.

An interesting problem left by this work is deciding if the moment condition can be removed.  Without an auxiliary moment condition we are able to establish the following weaker `in probability’ result.

Theorem Let be a sequence of independent, identically distributed mean zero random variables with finite variance . We then have that

Review of Public Key Encryption

Let us (informally) recall the definition of a public key cryptography system. Alice would like to send Bob a private message over an unsecured channel. Alice and Bob have never met before and we assume they do not share any secret information. Ideally, we would like a procedure where 1) Alice and Bob engage in a series of communications resulting in Bob learning the message 2) an eavesdropper, Eve, who intercepts all of the communications sent between Alice and Bob, should not learn any (nontrivial) information about the message . As stated, the problem is information theoretically impossible. However, this problem is classically solved under the heading of public key cryptography if we further assume that:

1) Eve has limited computational resources,
2) certain computational problems (such as factoring large integers or computing discrete logarithms in a finite group) are not efficiently solvable, and
3) we allow Alice and Bob to use randomization (and permit security to fail with very small probability).

More specifically, a public key protocol works as follows: Bob generates a private and public key, say and respectively. As indicated by the names, is publicly known but Bob retains as secret information. When Alice wishes to send a message to Bob she generates an encrypted ciphertext using the message , Bob’s public key and some randomness. She then sends this ciphertext to Bob via the public channel. When Bob receives the ciphertext he decrypts it using his secret key and recovers . While Eve has access to the ciphertext and the secret key , she is unable to learn any nontrivial information about the message (assuming our assumptions are sound). In fact, we require a bit more: even if this is repeated many times (with fixed keys), Eve’s ability to decrypt the ciphertext does not meaningfully improve.

Leakage Resilient Cryptography and our work

In practice, however, Eve may be able to learn information in addition to what she intercepts over Alice and Bob’s public communications via side channel attacks. Such attacks might include measuring the amount of time or energy Bob uses to carry out computations. The field of leakage resilient cryptography aims to incorporate protection against such attacks into the the security model. In this model, in addition to the ciphertext and public key, we let Eve select a (efficiently computable) function where is the bit length of and is a constant. We now assume, in addition to and , Eve  also gets to see . In other words, Eve gains a fair amount of information about the secret key, but not enough to fully determine it.

Moreover, we allow Eve to specify a different function every time Alice sends Bob a message. There is an obvious problem now, however. If the secret key remained static, then Eve could start by choosing to output the first bits, the second time she could choose to give the next bits, and if she carries on like this, after messages she would have recovered the entire secret key. To compensate for this we allow Bob to update his secret key between messages. The public key will remain the same.

There has been a lot of interesting work on this problem. In the works of Brakerski, Kalai, Katz, and Vaikuntanathan and Dodis, Haralambiev, Lopez-Alt, and Wichs many schemes are presented that are provably secure against continual leakage. In these schemes, however, information about the secret key is permitted to be leaked between updates, but only a tiny amount is allowed to be leaked during the update process itself.

In our current work, we offer the first scheme that allows a constant fraction of the information used in the update to be leaked. The proof is based on subgroup decision assumptions in composite order bilinear groups.

   Mathematicians, on the contrary, who may have the most perfect assurance, both of the truth and of the importance of their discoveries, are frequently very indifferent about the reception which they may meet with from the public. The two greatest mathematicians that I ever have had the honour to be known to, and, I believe, the two greatest that have lived in my time, Dr Robert Simpson of Glasgow, and Dr Matthew Stewart of Edinburgh, never seemed to feel even the slightest uneasiness from the neglect with which the ignorance of the public received some of their most valuable works. The great work of Sir Isaac Newton, his Mathematical Principles of Natural Philosophy, I have been told, was for several years neglected by the public. The tranquillity of that great man, it is probable, never suffered, upon that account, the interruption of a single quarter of an hour. Natural philosophers, in their independency upon the public opinion, approach nearly to mathematicians, and, in their judgments concerning the merit of their own discoveries and observations, enjoy some degree of the same security and tranquillity.

   The morals of those different classes of men of letters are, perhaps, sometimes somewhat affected by this very great difference in their situation with regard to the public.

   Mathematicians and natural philosophers, from their independency upon the public opinion, have little temptation to form themselves into factions and cabals, either for the support of their own reputation, or for the depression of that of their rivals. They are almost always men of the most amiable simplicity of manners, who live in good harmony with one another, are the friends of one another’s reputation, enter into no intrigue in order to secure the public applause, but are pleased when their works are approved of, without being either much vexed or very angry when they are neglected.

The entire text is available here.

Let denote a hypersurface in with surface measure . The restriction problem for is to determine for which pairs of does there exist an inequality of the form

We note that the left-hand side is not necessarily well-defined since we have restricted the function to the hypersurface , a set of measure zero in  . However, if we can establish this inequality for all Schwartz functions , then the operator that restricts to (denoted by ), can be defined whenever . In the Euclidean setting, the restriction problem has been extensively studied when is a sphere, paraboloid, and cone. In particular, it has been observed that restriction estimates are intimately connected to questions about certain partial differential equations as well as problems in geometric measure theory such as the Kakeya conjecture. The restriction conjecture states sufficient conditions on for the above inequality to hold. In the case of the sphere and paraboloid, the question is open in dimensions three and higher.

In 2002 Mockenhaupt and Tao initiated the study of the restriction phenomena in the finite field setting. Let us introduce some notation to formally define the problem in this setting. We let denote a finite field of characteristic . We let denote the unit circle in and define to be a non-principal character of . For example, when , we can set . We will be considering the vector space and its dual space . We can think of as endowed with the counting measure which assigns mass 1 to each point and as endowed with the normalized counting measure which assigns mass to each point (where denotes the size of , so the total mass is equal to 1 here).

For a complex-valued function on , we define its Fourier transform on by:

For a complex-valued function on , we define its inverse Fourier transform on by:

It is easy to verify that and .

We define the paraboloid as: . This is endowed with the normalized “surface measure” which assigns mass to each point in . We note that .
For a function , we define the function as follows:

For a complex-valued function on and , we define

For a complex-valued function on , we similarly define

Now we define a restriction inequality to be an inequality of the form

where denotes the best constant such that the above inequality holds. By duality, this is equivalent to the following extension estimate:

We will use the notation to denote that quantity is at most a constant times quantity , where this constant may depend on the dimension but not on the field size, . For a finite field , the constant will always be finite. The restriction problem in this setting is to determine for which can we upper bound independently of (i.e. for which does hold).

Mockenhaupt and Tao solved this problem for the paraboloid in two dimensions.  In three dimensions, we require not be a square in (without this restriction the parabaloid will contain non-trivial subspaces which lead to trivial counterexamples, but we will not elaborate on this here). For such , they showed that and for every . When , their bounds were polylogarithmic in . Mockenhaupt and Tao’s argument for the estimate proceeded by first establishing the estimate for characteristic functions. Here one can expand the norm and reduce the problem to combinatorial estimates. A well-known dyadic pigeonhole argument then allows one to pass back to general functions at the expense of a logarithmic power of . Following a similar approach (but requiring much more delicate Gauss sum estimates), Iosevich and Koh proved that and in higher dimensions (in odd dimensions some additional restrictions on are required). Again, however, this argument incurred a logarithmic loss at the endpoints from the dyadic pigeonhole argument.

In this note we remove the logarithmic losses mentioned above. Our argument begins by rewriting the norm as . We then adapt the arguments of the prior papers to the bilinear variant in the case that and are characteristic functions.

To obtain estimates for arbitrary functions , we can assume that is non-negative real-valued and decompose as a linear combination of characteristic functions, where the coefficients are negative powers of two (we can do this without loss of generality by adjusting only the constant of our bound). We can then employ the triangle inequality to upper bound by a double sum of terms like , where and are characteristic functions, weighted by negative powers of two. We then apply our bilinear estimate for characteristic functions to these inner terms and use standard bounds on sums to obtain the final estimates.

Our method yields the following theorems:

Theorem For the paraboloid in dimensions with not a square, we have and .
Theorem For the paraboloid in dimensions when is even or when is odd and for a prime congruent to 3 modulo 4 such that is not a multiple of 4, we have and .

We recently learned that in unpublished work Bennett, Carbery, Garrigos, and Wright have also obtained the results in the -dimensional case. Their argument proceeds rather differently than ours and it is unclear (at least to me) if their argument can be extended to the higher dimensional settings.

We say a set of natural numbers is sum-free if there is no solution to the equation with . The following is a well-known theorem of Erdős.

Theorem Let be a finite set of natural numbers. There exists a sum-free subset such that .

The proof of this theorem is a common example of the probabilistic method and appears in many textbooks. Alon and Kleitman have observed that Erdős’ argument essentially gives the theorem with the slightly stronger conclusion . Bourgain  has improved this further, showing that the conclusion can be strengthened to . Bourgain’s estimate is sharp for small sets, and improving it for larger sets seems to be a difficult problem. There has also been interest in establishing upper bounds for the problem. It seems likely that the constant cannot be replaced by a larger constant, however this is an open problem. In Erdős’ 1965 paper, he showed that the constant could not be replaced by a number greater than by considering the set . In 1990, Alon and Kleitman improved this to . In a recent survey of open problems in combinatorics, it is reported that Malouf has shown the constant cannot be greater than .  While we have not seen Malouf’s proof, we note that this can be established by considering the set . In this note we further improve on these results by showing that the optimal constant cannot be greater than .

Update (May 2, 2010): After posting this preprint, Stefan Neuwirth informed us that Rudin’s question had been previously answered by Y. Meyers in 1968. It appears that Meyers’ construction doesn’t, however, say anything about the anti-Freiman problem. Indeed Meyers’ set (and all of its subsets) contains a set of density . Hence, the construction of a set that doesn’t contain a large set still appears to be new. A revised version of the paper has been posted reflecting this information.  Most notably, we have changed the title to “On the Structure of Sets of Large Doubling”.

Allison Lewko and I recently arXiv’ed our paper “Sets of Large Doubling and a Question of Rudin“. The paper (1) answers a question of Rudin regarding the structure of sets (2) negatively answers a question of O’Bryant about the existence of a certain “anti-Freiman” theorem (3) establishes a variant of the (solved) Erdös-Newman conjecture. I’ll briefly describe each of these results below.

— Structure of sets —

Before describing the problem we will need some notation. Let and define to be the number of unordered solutions to the equation with . We say that is a set if for all . There is a similar concept with sums replaced by differences. Since this concept is harder to describe we will only introduce it in the case . For we define to be the number of solutions to the equation with . If for all nonzero we say that is a set.

Let be a subset of the integers , and call an -polynomial if it is a trigonometric polynomial whose Fourier coefficients are supported on (i.e. if ). We say that is a set (for ) if

holds for all -polynomials where the constant only depends on and . If is an even integer, we can expand out the norm in 1. This quickly leads to the following observation: If is a set then is also an set (, ). One can also easily show using the triangle inequality that the union of two sets is also a set. It follows that the finite union of sets is a set. In 1960 Rudin asked the following natural question: Is every set is a finite union of sets?

In this paper we show that the answer is no in the case of sets. In fact, we show a bit more than this. One can easily show that a set is also a set. Our first counterexample to Rudin’s question proceeded (essentially) by constructing a set which wasn’t the finite union of sets. This however raised the following variant of Rudin’s question: Is every set the mixed finite union of and sets? We show that the answer to this question is no as well. To do this we construct a set, A, which isn’t a finite union of sets, and a set, , which isn’t the finite union of sets. We then consider the product set which one can prove is a subset of . It isn’t hard to deduce from this that is a subset of that isn’t a mixed finite union of and sets. Moreover, one can (essentially) map this example back to while preserving all of the properties stated above. Generalizing this further, we show that there exists a set that doesn’t contain (in a sense that can be made precise) a large or . This should be compared with a related theorem of Pisier which states that every Sidon set contains a large independent set (it is conjectured that a Sidon set is a finite union of independent sets, however this is open).

We have been unable to extend these results to sets for . Very generally, part of the issue arises from the fact that the current constructions hinges on the existence of arbitrary large binary codes which can correct strictly more than a fraction of errors. To modify this construction (at least in a direct manner) to address the problem for, say, sets it appears one would need arbitrary large binary codes that can correct strictly more than a fraction of errors. However, one can show that such objects do not exist.

— Is there an anti-Freiman theorem? —

 Let be a finite set of integers and denote the sumset of as . A trivial inequality is the following

In fact, it isn’t hard to show that equality only occurs on the left if is an arithmetic progression and only occurs on the right if is a set. A celebrated theorem of Freiman states that if then is approximately an arithmetic progression. More precisely, if is a finite set satisfying for some constant , then is contained in a generalized arithmetic progression of dimension and size where and depend only on and not on .

It is natural to ask about the opposite extreme: if , what can one say about the structure of as a function only of ? A first attempt might be to guess that if for some positive constant , then can be decomposed into a union of sets where and depend only on . This is easily shown to be false. For example, one can start with a of elements contained in the interval and take its union with the arithmetic progression . It is easy to see that regardless of . However, the interval cannot be decomposed as the union of sets with and independent of .

There are two ways one might try to fix this problem: first, we might ask only that contains a set of size , where and depend only on . (This formulation was posed as an open problem by O’Bryant here). Second, we might ask that hold for all subsets for the same value of . Either of these changes would rule out the trivial counterexample given above. In this paper we show that even applying both of these modifications simultaneously is not enough to make the statement true. We provide a sequence of sets where holds for all of their subsets for the same value of , but if we try to locate a set, , of density in then must tend to infinity with the size of . As above, our initial construction of such a sequence of ‘s turned out to be sets. This leads us to the even weaker anti-Freiman conjecture:

(Weak Anti-Freiman) Suppose that satisfies and for all subsets . Then contains either a set or a set of size , where and depend only on .

We conclude by showing that even this weaker conjecture fails. The constructions are the same as those used in the results above. The two problems are connected by the elementary observation that if is a subset of a set then holds where only depends on the constant of the set .

— A variant of the Erdös-Newman conjecture —

In the early 1980′s Erdös and Newman independently made the following conjecture: For every there exists a that isn’t a finite union of sets for any . This conjecture was later confirmed by Erdös for certain values of using Ramsey theory, and finally resolved completely by Nešetřil and Rödl using Ramsey graphs. One further application of our technique is the following theorem which can be viewed as an analog of the Erdös-Newman problem with the roles of the union size and reversed.

Theorem 1 For every there exists a union of sets that isn’t a finite union of sets for any .

 More recently, simplified proofs of the dense model theorem have been obtained independently by Gowers and Reingold, Trevisan, Tulsiani and Vadhan. In addition, the latter group has found applications of these ideas in theoretical computer science. In this post we give an expository proof of the dense model theorem, substantially following the paper of Reingold, Trevisan, Tulsiani and Vadhan.   

 With the exception of the min-max theorem from game theory (which can be replaced by (or proved by) the Hahn-Banach theorem, as in Gowers’ approach) the presentation is self-contained.   

(We note that the the theorem, as presented below, isn’t explictly stated in the Green-Tao paper. Roughly speaking, these ideas can be used to simplify/replace sections 7 and 8 of that paper.)

 — The dense model theorem —   

 Let . We’ll primarily be interested in certain real-valued functions on the set . We define the expectation of a function, say , on to be and the inner product of to be . We will call a non-negative real-valued function, say , on a measure (This isn’t a measure in the analytic sense however this terminology has become standard in the literature on the subject) if .   

 We will call a function on , say , bounded if . Analogously a measure, say , will be called bounded if . For a fixed finite collection of bounded functions, , on , we say that two measure, say and , are -indistinguishable with respect to if for every . Furthermore, a measure on is said to be -pseudorandom with respect to if the measures and are -indistinguishable. Here denotes the characteristic function of the set .   

 In addition to the set we will also need to consider the larger class of functions . We can now state the dense model theorem.   

 Theorem 1 Fix and , a finite collection of bounded functions on . Furthermore, let be a -pseudorandom measure with respect to the set and a measure majorized by . There exists and (that depend\footnote{For the sake of simplicity we will not work out the dependency of these parameters on . We do however (very briefly) discuss the dependencies in the remark at the end of this section.} only on ) and a bounded measure such that and and are -indistinguishable with respect to .   

The thrust of the theorem is that and depend only on and not . At first the fact that is used in the hypothesis of the theorem and in the conclusion may seem strange. In applications, however, one often wishes to find a dense (-indistinguishable) model for a measure for a prescribed . One proceeds by locating , a majorant of , that is -indistinguishable from the measure . With applications of this form in mind, the statement of the theorem may seem more natural.   

 We will split the proof of the theorem into several parts/lemmas. Throughout will denote the set of bounded measures of expectation . We’ll typically denote an element of with the symbol .   

 Lemma 2 Let and denote the convex hull of . Furthermore let be a real-valued polynomial (depending only on ) that maps to . If there is a function of the form with that -distinguishes from then there exists a function that -distinguishes from .   

Proof: We note that and are -distinguishable with respect to if and only if they are -distinguishable with respect to (this allows us to remove the absolute value from the definition of -distinguishability given above).   

Next we note that it suffices to show that and are -distinguishable with respect to , the convex hull of . To see this assume that -distinguishes and with , , and . We then have that , which easily implies that for some .   

 Furthermore, let be a real-valued polynomial that depends only on . We claim that it then suffices to show that there exists a function such that -distinguishes and . To see this set equal to the magnitude of the largest coefficient of and the degree of . Letting we have that hence   

 and thus, for some ,   

 Since the right-hand side depends only on the proof is complete.    

 It now suffices to assume that the conclusion of the the theorem is false and find a function of the form with that -distinguishes from , which would provide a contradiction.   

 Lemma 3 Assume that for every there exists a such that (in other words, assume the theorem is false). Then there exists a function that -distinguishes every function from . This is to say   

for every .    

Proof: Let denote a finite set\footnote{To see that such a set exists consider .} of bounded measures on such that the convex hull of is . Consider the matrix with entries . By the min-max theorem there exists and such that for all and for every .   

 By the hypothesis of the theorem we have that for every there exists such that . Thus there exists a such that . Hence and taking completes the proof.    

 We now let be a set of elements of that maximizes the quantity . Additionally let denote an element of the set that maximizes . Define . By construction we have that thus and . This implies that -distinguishes from .   

 Lemma 4 Let where . Then there exists a threshold such that .    

Proof: We have previously observed that . Using the fact that we have that . Combining this with the observation that we have that  

 Assuming that the conclusion of the lemma is false, that is , we can conclude    

This would, however, contradict the inequality   

 which is an easy consequence of the inequality derived in the second sentence of this proof. Hence the proof is complete.    

 Lemma 5 We have that for every .   

Proof: Recall that . If for any then must vanish identically on . However this implies that    

 For large the term is negligible and thus   

 But this would contradict the conclusion of Lemma 4.    

 Let us briefly summarize the strategy for completing the proof of the theorem. From the previous lemma we have . However, since distinguishes and it must also distinguish and . This would contradict the hypothesis of the theorem if . In light of Lemma 2 it then suffices to show that can be approximated by a function of the form where . For this purpose, let be a polynomial mapping into and satisfying   for and for .  (The existence of such a polynomial can be obtained from standard variants of Weierstrass’ approximation theorem.)   

Lemma 6 Let be as defined above. Then -distinguishes from .   

Proof: From the definition of we have that 

Using the lower bound in (1), the pointwise inequality , Lemma 4 and Lemma 5 (in this order) we have that  

Using the upper bound from (1) we further have that  

Putting these two calculations together we conclude that  

which completes the proof.  

Recalling that we may apply Lemma 2 to complete the proof of the dense model theorem.   

 Remark 1 We note that the only step of the proof where we haven’t explicitly recorded the relationship between , and is in the dependency of the polynomial on . Using a quantitative form of Weierstrass’ approximation theorem one can obtain and . We refer the reader here for details.

Updated 12/17/2009: Fixed formatting in proof of Lemma 6.

We named the dog (who was found without tags or a microchip) after former UT mathematician George Halstead (who also had some trouble making a home in Austin). Our efforts were first met by several failed applications. While these were disappointing, they did buy Halstead an extended stay at Town Lake.

I just got off the phone with Town Lake and I am happy to report that Halstead has been adopted! She will leave for her forever home tomorrow. Many thanks to everyone who helped make this happen!

Unfortunately many of the pets at Town Lake will never get the second chance that Halstead will (although Austin Pets Alive! has been making great strides to change this). I encourage anyone considering adopting a pet to first visit Town Lake (or your local animal shelter).

Update 11/23/2009: Town Lake confirms that Halstead did leave the shelter yesterday with her new family.

for some positive constant . This is stronger than the observation (which was made much earlier) that almost every positive integer can be expressed as the sum of two primes. In this post we’ll be interested in sets of integers with the property that most integers can be expressed as the sum of two elements from the set. To be more precise we’ll say that a set of positive integers has the Goldbach property (GP) if the sumset consists of a positive proportion of the integers. From the preceding discussion we have that the set of primes has the GP. (This discussion is closely related to the theory of thin bases.)

A natural first question in investigating such sets would be to ask how thin such a set can be. Simply considering the number of possible distinct sums, the reader can easily verify that a set (of positive integers) with the GP must satisfy

This is to say that a set of positive integers with the GP must satisfy for all large . Recall that the prime number theorem gives us that  for the set of primes . Thus the primes are much thicker than a set with the GP needs to be (at least from naive combinatorial considerations).

Considering this, one might ask if there is a subset of the primes with the GP but having significantly lower density in the integers. I recently (re)discovered that the answer to this question is yes. In particular we have that

 Theorem 1 There exists a subset of the primes such that has positive density in the integers and

 This last inequality shows that is as thin as such a set can be. In fact, one can show that a generic subset of the primes of this density will work. This result, in an more quantitative form, has been observed by Granville and earlier by Wirsing. Here I’ll present the somewhat different argument I found which is based on a theorem of Bourgain.

 Some properties of sets

In this section we define sets and record some of their properties. Let . For we say that is a set (with constant ) if for every function such that (here denotes the -th Fourier coefficient of ) we have

There is a rich theory to sets (being an instance of the Restriction phenomenon in Fourier analysis), however we will only need a few key properties. The following theorem was proved in this paper of Rudin (which is also where sets were first defined).

Theorem 2 If is a set for , and is an arithmetic progression then .  More specifically, we have that

This tells us, for example, that a set must satisfy for large . These sets also have a number of nice combinatorial properties. Here we record a basic lemma (which can be found as Lemma 4.30 in Tao and Vu’s Additive Combinatorics) which is easily deduced from Parseval’s identity.

Lemma 3 Let be a set with constant and let be a finite subset of . We then have

 We now state a theorem of Bourgain.

 Theorem 4 For any there exists a set, consisting of only of primes such that

 The proof of this theorem is fairly involved and relies on number theoretic as well as probabilistic arguments. We will only need this result in the case when . In this case (since is an even integer and norms can be expanded) both parts of the argument can be significantly simplified. In particular the only number theoretic information that is required are upper bounds on the number of ways an integer can be represented as the sum of two primes (see, for example, this paper of Green and Tao). Information of this sort can be readily obtained by standard sieve methods.

Let us now fit these pieces together to prove the theorem.

Proof of Theorem 1

Let be the set given by Bourgain’s theorem. By virtue of the fact that is a set we have that 

On the other hand, for sufficiently large we also have that . Applying Lemma 3, we have that . We can conclude that 

and the theorem follows.

Some Remarks

The argument here (via the proof of Bourgain’s theorem) constructs the set in a probabilistic manner. This is also the case in the arguments of Granville and Wirsing. In fact all of these arguments proceed by showing that a generic (in an appropriate sense) has the property that one is seeking. It would be very interesting to construct an explicit example of such a set.

As we remarked earlier we are only using Bourgain’s theorem when is an even integer, in which case his argument can be simplified considerably. It would be interesting to find an analogous combinatorial interpretation of sets for . Presumably this would lead to a deeper result about the primes.

The proof of Bourgain’s theorem uses very little about the primes. Similar ideas can be applied to other number theoretic sets, such as polynomial sequences. In fact, his paper gives a general criterion for a set to containing a subsets of maximal density.