Lewko's blog

An Exact Asymptotic for the Square Variation of Partial Sum Processes

Posted in math.PR, Paper by Mark Lewko on June 6, 2011

Allison and I just arxiv’ed our paper An Exact Asymptotic for the Square Variation of Partial Sum Processes.

Let \{X_{i}\} be a sequence of independent, identically distributed random variables with mean \mu < \infty . The strong law of large numbers asserts that

\sum_{i=1}^{N}X_{i} \sim N\mu

almost surely. Without loss of generality, one can assume that X_{i} are mean-zero by defining Y_{i}=X_{i}-\mu. If we further assume a finite variance, that is \mathbb{E}\left[|X_{i}|^2 \right] = \sigma^2 < \infty, the Hartman-Wintner  law of the iterated logarithm gives an exact error estimate for the strong law of large numbers. More precisely,

\left|\sum_{i=1}^{N} X_{i} \right|^2\leq (2+o(1))\sigma^2 N \ln\ln (N)

where the constant 2 can not be replaced by a smaller constant. That is, the quantity \sum_{i=1}^{N}X_{i} gets as large/small as \pm \sqrt{ (2-\epsilon) \sigma N \ln\ln (N)} 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 \{X_{i}\} be a sequence of independent, identically distributed mean zero random variables with variance \sigma and satisfying \mathbb{E}\left[|X_{i}|^{2+\delta}\right] < \infty.  If we let \mathcal{P}_{N} denote the set of all possible partitions of the interval [N] into subintervals, then we have almost surely:

 \max_{\pi \in \mathcal{P}_{N}} \sum_{I \in \pi } | \sum_{i\in I} X_{i}|^2 \sim 2 \sigma^2N \ln \ln(N).

Choosing the partition \pi, to contain a single interval  J=[1,N] 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 \mathbb{E}\left[|X_{i}|^{2+\delta}\right] < \infty can be removed.  Without an auxiliary moment condition we are able to establish the following weaker `in probability’ result.

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

\frac{\max_{\pi \in \mathcal{P}_{N}} \sum_{I \in \pi } | \sum_{i\in I} X_{i}|^2}{2 \sigma^2 N \ln \ln(N)} \xrightarrow{p} 1

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How to leak on key updates

Posted in Cryptography, Paper by Mark Lewko on February 3, 2011

Leakage resilient cryptography is an exciting area of cryptography that aims to build cryptosystems that provide security against side channel attacks. In this post I will give a nontechnical description of a common leakage resilient security model, as well as describe a recent paper in the area with Allison Lewko and Brent Waters, titled “How to Leak on Key Updates”.

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 M 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 M 2) an eavesdropper, Eve, who intercepts all of the communications sent between Alice and Bob, should not learn any (nontrivial) information about the message M. 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 SK and PK respectively. As indicated by the names, PK is publicly known but Bob retains SK as secret information. When Alice wishes to send a message M to Bob she generates an encrypted ciphertext C using the message M, Bob’s public key PK 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 SK and recovers M. While Eve has access to the ciphertext C and the secret key SK, she is unable to learn any nontrivial information about the message M (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 F:\{0,1\}^{\ell}\rightarrow\{0,1\}^{\mu \ell} where \ell is the bit length of SK and 0<\mu<1 is a constant. We now assume, in addition to C and PK, Eve  also gets to see F(SK). 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 F every time Alice sends Bob a message. There is an obvious problem now, however. If the secret key SK remained static, then Eve could start by choosing F to output the first \mu \ell bits, the second time she could choose F to give the next \mu \ell bits, and if she carries on like this, after 1/\mu 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.

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Restriction estimates for the paraboloid over finite fields

Posted in Fourier Analysis, math.CA, Paper by Mark Lewko on September 19, 2010

Allison and I recently completed a paper titled Restriction estimates for the paraboloid over finite fields. In this note we obtain some endpoint restriction estimates for the paraboloid over finite fields.

Let S denote a hypersurface in \mathbb{R}^{n} with surface measure d\sigma. The restriction problem for S is to determine for which pairs of (p,q) does there exist an inequality of the form

\displaystyle ||\hat{f}||_{L^{p'}(S,d\sigma)} \leq C ||f||_{L^{q'}(\mathbb{R}^n)}.

We note that the left-hand side is not necessarily well-defined since we have restricted the function \hat{f} to the hypersurface S, a set of measure zero in  \mathbb{R}^{n}. However, if we can establish this inequality for all Schwartz functions f, then the operator that restricts \hat{f} to S (denoted by \hat{f}|_{S}), can be defined whenever f \in L^{q}. In the Euclidean setting, the restriction problem has been extensively studied when S 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 (p,q) 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 F denote a finite field of characteristic p >2. We let S^{1} denote the unit circle in \mathbb{C} and define e: F \rightarrow S^1 to be a non-principal character of F. For example, when F = \mathbb{Z}/p \mathbb{Z}, we can set e(x) := e^{2\pi i x/p}. We will be considering the vector space F^n and its dual space F_*^n. We can think of F^n as endowed with the counting measure dx which assigns mass 1 to each point and F_*^n as endowed with the normalized counting measure d\xi which assigns mass |F|^{-n} to each point (where |F| denotes the size of F, so the total mass is equal to 1 here).

For a complex-valued function f on F^n, we define its Fourier transform \hat{f} on F_*^n by:

\displaystyle \hat{f}(\xi) := \sum_{x \in F^n} f(x) e(-x \cdot \xi).

For a complex-valued function g on F_*^n, we define its inverse Fourier transform g^{\vee} on F^n by:

\displaystyle g^{\vee}(x) := \frac{1}{|F|^n} \sum_{\xi \in F_*^n} g(\xi) e(x\cdot \xi).

It is easy to verify that (\hat{f})^\vee = f and \widehat{(g^{\vee})} = g.

We define the paraboloid \mathcal{P} \subset F_*^n as: \mathcal{P} := \{(\gamma, \gamma \cdot \gamma): \gamma \in F_*^{n-1}\}. This is endowed with the normalized “surface measure” d\sigma which assigns mass |\mathcal{P}|^{-1} to each point in \mathcal{P}. We note that |\mathcal{P}| = |F|^{n-1}.
For a function f: \mathcal{P} \rightarrow \mathbb{C}, we define the function (f d\sigma)^\vee: F^n \rightarrow \mathbb{C} as follows:

\displaystyle (f d\sigma)^\vee (x) := \frac{1}{|\mathcal{P}|} \sum_{\xi \in \mathcal{P}} f(\xi) e(x \cdot \xi).

For a complex-valued function f on F^n and q \in [1, \infty), we define

\displaystyle ||f||_{L^q(F^n, dx)} := \left( \sum_{x \in F^n} |f(x)|^q \right)^{\frac{1}{q}}.

For a complex-valued function f on \mathcal{P}, we similarly define

\displaystyle ||f||_{L^q(\mathcal{P},d\sigma)} := \left( \frac{1}{|\mathcal{P}|} \sum_{\xi \in \mathcal{P}} |f(\xi)|^q \right)^{\frac{1}{q}}.

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

\displaystyle ||\hat{f}||_{L^{p'}(S,d\sigma)} \leq \mathcal{R}(p\rightarrow q) ||f||_{L^{q'}(\mathbb{R}^n)},

where \mathcal{R}(p\rightarrow q) denotes the best constant such that the above inequality holds. By duality, this is equivalent to the following extension estimate:

||(f d\sigma)^\vee||_{L^q(F^n, dx)} \leq \mathcal{R}(p\rightarrow q) ||f||_{L^p(\mathcal{P},d\sigma)}.

We will use the notation X \ll Y to denote that quantity X is at most a constant times quantity Y, where this constant may depend on the dimension n but not on the field size, |F|. For a finite field F, the constant \mathcal{R}(p\rightarrow q) will always be finite. The restriction problem in this setting is to determine for which (p,q) can we upper bound \mathcal{R}(p\rightarrow q) independently of |F| (i.e. for which (p,q) does \mathcal{R}(p \rightarrow q) \ll 1 hold).

Mockenhaupt and Tao solved this problem for the paraboloid in two dimensions.  In three dimensions, we require -1 not be a square in F (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 F, they showed that \mathcal{R}(8/5+\epsilon \rightarrow 4) \ll 1 and \mathcal{R}(2 \rightarrow \frac{18}{5}+\epsilon) \ll 1 for every \epsilon>0. When \epsilon=0, their bounds were polylogarithmic in |F|. Mockenhaupt and Tao’s argument for the \mathcal{R}(8/5 \rightarrow 4) estimate proceeded by first establishing the estimate for characteristic functions. Here one can expand the L^4 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 |F|. Following a similar approach (but requiring much more delicate Gauss sum estimates), Iosevich and Koh proved that \mathcal{R}(\frac{4n}{3n-2}+ \epsilon \rightarrow 4) \ll 1 and \mathcal{R}(2 \rightarrow \frac{2n^2}{n^2-2n+2} + \epsilon) \ll 1 in higher dimensions (in odd dimensions some additional restrictions on F 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 L^4 norm as ||(fd\sigma)^{\vee}||_{L^4}=||(fd\sigma)^{\vee}(fd\sigma)^{\vee}||_{L^2}^{1/2}. We then adapt the arguments of the prior papers to the bilinear variant ||(fd\sigma)^{\vee}(gd\sigma)^{\vee}||_{L^2}^{1/2} in the case that f and g are characteristic functions.

To obtain estimates for arbitrary functions f, we can assume that f is non-negative real-valued and decompose f 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 ||(fd\sigma)^{\vee}||_{L^4} by a double sum of terms like ||(\chi_j d\sigma)^{\vee}(\chi_k d\sigma)^{\vee}||_{L^2}^{1/2}, where \chi_i and \chi_j 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 3 dimensions with -1 not a square, we have \mathcal{R}(8/5 \rightarrow 4) \ll 1 and \mathcal{R}(2 \rightarrow \frac{18}{5}) \ll 1.
Theorem For the paraboloid in n dimensions when n \geq 4 is even or when n is odd and |F| = q^m for a prime q congruent to 3 modulo 4 such that m(n-1) is not a multiple of 4, we have \mathcal{R}(\frac{4n}{3n-2} \rightarrow 4) \ll 1 and \mathcal{R}(2 \rightarrow \frac{2n^2}{n^2-2n+2}) \ll 1.

We recently learned that in unpublished work Bennett, Carbery, Garrigos, and Wright have also obtained the results in the 3-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.

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An improved upper bound for the sum-free subset constant

Posted in math.CO, Paper, Uncategorized by Mark Lewko on August 27, 2010

I recently arXiv’ed a short note titled An Improved Upper Bound for the Sum-free Subset Constant. In this post I will briefly describe the result.

We say a set of natural numbers A is sum-free if there is no solution to the equation x+y=z with x,y,z \in A. The following is a well-known theorem of Erdős.

Theorem Let A be a finite set of natural numbers. There exists a sum-free subset S \subseteq A such that |S| \geq \frac{1}{3}|A|.

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 |S| \geq \frac{|A|+1}{3}. Bourgain  has improved this further, showing that the conclusion can be strengthened to |S| \geq \frac{|A| + 2}{3}. 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 1/3 cannot be replaced by a larger constant, however this is an open problem. In Erdős’ 1965 paper, he showed that the constant \frac{1}{3} could not be replaced by a number greater than 3/7 \approx .429 by considering the set \{2,3,4,5,6,8,10\}. In 1990, Alon and Kleitman improved this to 12/29 \approx .414. In a recent survey of open problems in combinatorics, it is reported that Malouf has shown the constant cannot be greater than 4/10 = .4.  While we have not seen Malouf’s proof, we note that this can be established by considering the set \{1,2,3,4,5,6,8,9,10,18\}. In this note we further improve on these results by showing that the optimal constant cannot be greater than 11/28 \approx .393.

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Sets of large doubling and a question of Rudin

Posted in Fourier Analysis, math.CA, math.CO, Paper by Mark Lewko on April 2, 2010

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 B_{2}[2] set of density 1/4. Hence, the construction of a \Lambda(4) set that doesn’t contain a large B_{2}[2] 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 {\Lambda(4)} 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 {\Lambda(4)} sets —

Before describing the problem we will need some notation. Let {S \subset {\mathbb Z}^d} and define {R_{h}(n)} to be the number of unordered solutions to the equation {x_{1}+\ldots + x_{h}=n} with {x_{1},\ldots,x_{h} \in S}. We say that {S} is a {B_{h}[G]} set if {R_{h}(n) \leq G} for all {n \in Z^d}. 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 {h=2}. For {S \subset Z^{d}} we define {R_{2}^{\circ}(n)} to be the number of solutions to the equation {x_{1}-x_{2} = n} with {x_{1},x_{2}\in S}. If {R_{2}^{\circ}(n)\leq G} for all nonzero {n} we say that {S} is a {B_{2}^{\circ}[G]} set.

Let {S} be a subset of the integers {{\mathbb Z}^{d}}, and call {f : \mathbb{T}^{d} \rightarrow {\mathbb C}} an {S}-polynomial if it is a trigonometric polynomial whose Fourier coefficients are supported on {S} (i.e. {\hat{f}(n) = 0} if {n \in {\mathbb Z^{d}} \setminus S}). We say that {S} is a {\Lambda(p)} set (for {p>2}) if

\displaystyle ||f||_{L^p} \leq K_{p}(S) ||f||_{L^{2}} \ \ \ \ \ (1)

holds for all {S}-polynomials where the constant {K_{p}(S)} only depends on {S} and {p}. If {p} is an even integer, we can expand out the {L^{p}} norm in 1. This quickly leads to the following observation: If {S} is a {B_{h}[G]} set then {S} is also an {\Lambda(2h)} set ({h>1}, {h \in Z}). One can also easily show using the triangle inequality that the union of two {\Lambda(p)} sets is also a {\Lambda(p)} set. It follows that the finite union of {B_{h}[G]} sets is a {\Lambda(2h)} set. In 1960 Rudin asked the following natural question: Is every {\Lambda(2h)} set is a finite union of {B_{h}[G]} sets?

In this paper we show that the answer is no in the case of {\Lambda(4)} sets. In fact, we show a bit more than this. One can easily show that a {B_{2}^{\circ}[G]} set is also a {\Lambda(4)} set. Our first counterexample to Rudin’s question proceeded (essentially) by constructing a {B_{2}^{\circ}[2]} set which wasn’t the finite union of {B_{2}[G]} sets. This however raised the following variant of Rudin’s question: Is every {\Lambda(4)} set the mixed finite union of {B_{2}[G]} and {B_{2}^{\circ}[G]} sets? We show that the answer to this question is no as well. To do this we construct a {B_{2}[G]} set, A, which isn’t a finite union of {B_{2}^{\circ}[G]} sets, and a {B_{2}^{\circ}[G]} set, {B}, which isn’t the finite union of {B_{2}[G]} sets. We then consider the product set {S= A \times B \subset Z^{2}} which one can prove is a {\Lambda(4)} subset of {Z^{2}}. It isn’t hard to deduce from this that {S} is a {\Lambda(4)} subset of {Z^2} that isn’t a mixed finite union of {B_{2}[G]} and {B_{2}^{\circ}[G]} sets. Moreover, one can (essentially) map this example back to {Z} while preserving all of the properties stated above. Generalizing this further, we show that there exists a {\Lambda(4)} set that doesn’t contain (in a sense that can be made precise) a large {B_{2}[G]} or {B_{2}^{\circ}[G]}. 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 {\Lambda(2h)} sets for {h>2}. 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 {1/2} fraction of errors. To modify this construction (at least in a direct manner) to address the problem for, say, {\Lambda(6)} sets it appears one would need arbitrary large binary codes that can correct strictly more than a {2/3} fraction of errors. However, one can show that such objects do not exist.

— Is there an anti-Freiman theorem? —

 Let {A} be a finite set of integers and denote the sumset of {A} as {A+A = \{a+b : a,b \in A\}}. A trivial inequality is the following

\displaystyle 2|A|-1 \leq |A+A| \leq {|A| \choose 2}.

In fact, it isn’t hard to show that equality only occurs on the left if {A} is an arithmetic progression and only occurs on the right if {A} is a {B_{2}[1]} set. A celebrated theorem of Freiman states that if {|A+A| \approx |A|} then {A} is approximately an arithmetic progression. More precisely, if {A} is a finite set {A \subseteq {\mathbb Z}} satisfying {|A+A| \leq \delta |A|} for some constant {\delta}, then {A} is contained in a generalized arithmetic progression of dimension {d} and size {c |A|} where {c} and {d} depend only on {\delta} and not on {|A|}.

It is natural to ask about the opposite extreme: if {|A+A| \geq \delta |A|^2}, what can one say about the structure of {A} as a function only of {\delta}? A first attempt might be to guess that if {|A+A|\geq \delta |A|^2} for some positive constant {\delta}, then {A} can be decomposed into a union of {k} {B_2[G]} sets where {k} and {G} depend only on {\delta}. This is easily shown to be false. For example, one can start with a {B_2[1]} of {n} elements contained in the interval {[n+1,\infty)} and take its union with the arithmetic progression {[1,n]}. It is easy to see that {|A+A| \geq \frac{1}{10} |A|^2} regardless of {n}. However, the interval {[1,n]} cannot be decomposed as the union of {k} {B_2[G]} sets with {k} and {G} independent of {n}.

There are two ways one might try to fix this problem: first, we might ask only that {A} contains a {B_2[G]} set of size {\delta' |A|}, where {\delta'} and {G} depend only on {\delta}. (This formulation was posed as an open problem by O’Bryant here). Second, we might ask that {|A'+A'|\geq \delta |A'|^2} hold for all subsets {A' \subseteq A} for the same value of {\delta}. 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 {A \subseteq {\mathbb Z}} where {|A'+A'|\geq \delta |A'|^2} holds for all of their subsets for the same value of {\delta}, but if we try to locate a {B_2[G]} set, {B}, of density {1/k} in {A} then {k} must tend to infinity with the size of {A}. As above, our initial construction of such a sequence of {A}‘s turned out to be {B^\circ_2[2]} sets. This leads us to the even weaker anti-Freiman conjecture:

(Weak Anti-Freiman) Suppose that {A \subseteq {\mathbb Z}} satisfies {|A'+A'|\geq \delta |A'|^2} and {|A'-A'|\geq \delta |A'|^2} for all subsets {A' \subseteq A}. Then {A} contains either a {B_2[G]} set or a {B^\circ_2[G]} set of size {\geq \delta' |A|}, where {G} and {\delta'} depend only on {\delta}.

We conclude by showing that even this weaker conjecture fails. The constructions are the same as those used in the {\Lambda(4)} results above. The two problems are connected by the elementary observation that if {A'} is a subset of a {\Lambda(4)} set {A} then {|A'+A'|\geq \delta |A'|^2} holds where {\delta} only depends on the {\Lambda(4)} constant {K_{4}(A)} of the set {A}.

— A variant of the Erdös-Newman conjecture —

In the early 1980′s Erdös and Newman independently made the following conjecture: For every {G} there exists a {B_{2}[G]} that isn’t a finite union of {B_{2}[G']} sets for any {G'\leq G-1}. This conjecture was later confirmed by Erdös for certain values of {G} 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 {G} reversed.

Theorem 1 For every {k >1} there exists a union of k {B_{2}[1]} sets that isn’t a finite union of {k'\leq k-1} {B_{2}[G]} sets for any {G}.

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