A counterexample to a strong variant of the Polynomial Freiman-Ruzsa conjecture in Euclidean space
DD ISCRETE A NALYSIS , 2017:8, 6 pp.
A Counterexample to a Strong Variant of thePolynomial Freiman-Ruzsa Conjecture inEuclidean Space
Shachar Lovett ∗ Oded Regev † Received 3 February 2017; Published 9 May 2017
Abstract:
The Polynomial Freiman-Ruzsa conjecture is one of the central open problems inadditive combinatorics. If true, it would give tight quantitative bounds relating combinatorialand algebraic notions of approximate subgroups. In this note, we restrict our attention tosubsets of Euclidean space. In this regime, the original conjecture considers approximatealgebraic subgroups as the set of lattice points in a convex body. Green asked in 2007 whetherthis can be simplified to a generalized arithmetic progression, while not losing more than apolynomial factor in the underlying parameters. We give a negative answer to this question,based on a recent reverse Minkowski theorem combined with estimates for random lattices.
The Polynomial Freiman-Ruzsa conjecture, first suggested by Katalin Marton, would, if true, give apolynomial relation between combinatorial and algebraic notions of approximate groups. In this note, werestrict our attention to subsets of Euclidean space.Let A be a finite subset of R n . Its Minkowski sumset is A + A = { a + a : a , a ∈ A } . The doublingfactor of A is | A + A | / | A | . Sets of small doubling can be viewed as a combinatorial notion of “approximate ∗ Supported by an NSF CAREER award 1350481, an NSF CCF award 1614023, and a Sloan fellowship. † Supported by the Simons Collaboration on Algorithms and Geometry and by the National Science Foundation (NSF) underGrant No. CCF-1320188. Any opinions, findings, and conclusions or recommendations expressed in this material are those ofthe authors and do not necessarily reflect the views of the NSF. c (cid:13) cb Licensed under a Creative Commons Attribution License (CC-BY) DOI: 10.19086/da.1640 a r X i v : . [ m a t h . N T ] M a y HACHAR L OVETT AND O DED R EGEV subgroups.” In R n , a natural example is that of lattice points in a symmetric convex body, as the followingstandard fact shows. Claim 1.1.
Let B ⊂ R n be a symmetric (i.e., B = − B) convex body, let L ⊂ R n be an n-dimensionallattice, and take A = L ∩ B. Then | A + A | ≤ n | A | .Proof. Note that A + A ⊆ L ∩ B . Next, let D be a maximal subset of L ∩ B satisfying that the sets in { x + B / x ∈ D } are disjoint. Then on one hand, by the maximality of D , L ∩ B ⊆ D + ( L ∩ B ) . On theother hand, by a volume packing argument, | D | ≤ n . Therefore, | A + A | ≤ | L ∩ B | ≤ | D | · | A | ≤ n | A | . We remark that Claim 1.1 fails for non-symmetric convex bodies. Take, for instance, B ⊂ R to be theconvex hull of the four points ( N , , ) , ( − N , , ) , ( , N , ) , ( , − N , ) for some large integer N , L = Z and set A = L ∩ B . Then | A | = N + | A + A | ≥ ( N + ) .Freiman [2] showed that sets of small doubling must be contained in a low-dimensional affinesubspace. Concretely, if | A + A | ≤ K | A | then A is supported on a subspace of dimension 2 K . However,there is an exponential gap between this bound (which is tight) and the example of lattice points in convexbodies. The Polynomial Freiman-Ruzsa (PFR) conjecture is an attempt to bridge this gap. One naturalformulation is the following (see [3] for a further discussion). Conjecture 1.2 (PFR conjecture in R n ) . There exists an absolute constant c > such that the followingholds. Let A ⊂ R n be a set with | A + A | ≤ K | A | . Then for some d ≤ c log K there exist1. a d-dimensional lattice L ⊂ R d ,
2. a symmetric convex body B ⊂ R d ,3. a linear map ϕ : R d → R n , and4. a set X ⊂ R n of size | X | ≤ K c such that | L ∩ B | ≤ | A | and A ⊂ ϕ ( L ∩ B ) + X .
Green [3] asked (and speculated that the answer is negative) whether Conjecture 1.2 could potentiallybe strengthened, where ϕ ( L ∩ B ) is replaced by a more restricted object, a generalized arithmeticprogression (GAP). A d -dimensional GAP G ⊂ R n is a set of the form G = (cid:110) x + d ∑ i = α i x i : a i ≤ α i ≤ b i , α i ∈ Z (cid:111) , (1)where x , x , . . . , x d ∈ R n and a , . . . , a d , b , . . . , b d ∈ Z . By a slight abuse of notation, the size of a GAP G is its size as a multiset, namely | G | = ∏ di = ( b i − a i + ) .As one can observe, a GAP G can be written as ϕ ( L ∩ B ) + x as defined in Conjecture 1.2. Moreprecisely, assuming that b i − a i is even for all i , we can take L = Z d , B = ∏ di = [ − ( b i − a i ) / , ( b i − a i ) / ] , ϕ ( e i ) = x i where e , . . . , e d are the standard unit vectors in R d , and x = x + ∑ i ( a i + b i ) x i /
2. Moreover,GAPs have a simpler combinatorial structure than the general case of linear images of lattice points in aconvex body. As such, it will be pleasing if Conjecture 1.2 can be simplified as follows. One can restrict without loss of generality to L = Z d . D ISCRETE A NALYSIS , 2017:8, 6pp. 2 C
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Conjecture 1.3 (PFR conjecture in R n ; GAP version) . There exists an absolute constant c > such thatthe following holds. Let A ⊂ R n be a set with | A + A | ≤ K | A | . Then there exist1. a d-dimensional GAP G ⊂ R n of dimension d ≤ c log K;2. a set X ⊂ R n of size | X | ≤ K c ,such that | G | ≤ | A | and A ⊂ G + X .
In this note we refute Conjecture 1.3. We show that the bound on | X | cannot be polynomial in K —itis at least of the order of K c (cid:48) loglog K for some c (cid:48) >
0. We note that our example is of the form L ∩ B asdefined in Conjecture 1.2, so it does not shed new light on the original conjecture.Let A , G , X be as in Conjecture 1.3. By an averaging argument, there exists an x ∈ X such that | A ∩ ( G + x ) | ≥ K − c | A | . Note that G + x is also a GAP. Thus, the following theorem is sufficient to rule out Conjecture 1.3. Hereand below, by a “random n -dimensional lattice” we mean a lattice chosen from the unique probabilitymeasure over the set of determinant-one lattices in R n that is invariant under SL n ( R ) [7]. Theorem 1.4.
For any c ≥ the following holds. Let B ⊂ R n be a Euclidean ball of radius n / and L ⊂ R n be a random n-dimensional lattice. Set A = L ∩ B, and recall from Claim 1.1 that its doublingfactor satisfies K ≤ n . Then with probability tending to as n → ∞ over the choice of L , the followingholds. For any d-dimensional GAP G ⊂ R n with d ≤ cn and | G | ≤ | A | , | A ∩ G | ≤ n − n / ( c ) | A | ≤ K − ( / c ) loglog K | A | . We note that the bound on the intersection of lattice points in a convex body with a GAP in Theo-rem 1.4 is tight, up to the constant factors. Lemma 3.33 in [8] shows that if A = L ∩ B where L is a latticeand B is any symmetric convex body in R n , then there exists an n -dimensional GAP G ⊂ A such that | A ∩ G | ≥ ( c (cid:48)(cid:48) n ) − n / | A | for some constant c (cid:48)(cid:48) > A rank- d lattice L ⊂ R n is the set of integer linear combinations of d linearly independent vectors B = ( b , . . . , b d ) , L = (cid:110) d ∑ i = a i b i : a i ∈ Z (cid:111) . The determinant of the lattice is given by det ( L ) = det ( B t B ) / , where we view B as an n × d matrix. Onecan verify that the determinant is independent of the choice of basis of a lattice. We say that a subspace W ⊂ R n is a lattice subspace of L if it is spanned by vectors in the lattice L . We denote by B n ( r ) theEuclidean ball of radius r in R n centered at the origin. D ISCRETE A NALYSIS , 2017:8, 6pp. 3
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We will use the following lower bound on the determinants of sublattices of a random lattice. Theformulation below is due to [6], which in turn is based on the estimates of [9].
Theorem 3.1 ([6, Proposition 3]) . Let L be a random n-dimensional lattice. Then with probabilitytending to as n → ∞ , it holds that for any lattice subspace W of L , det ( L ∩ W ) / dim ( W ) ≥ c · n ( − dim ( W ) / n ) / , where c > denotes an absolute constant. We will also need the following “reverse Minkowski” theorem, earlier conjectured by Dadush [1].
Theorem 3.2 ([5]) . Let n ≥ and L ⊂ R n be a lattice satisfying that for any lattice subspace W of L , det ( L ∩ W ) / dim ( W ) ≥ R. Then, for any r ≥ R, | L ∩ B n ( r ) | ≤ ( / ) exp ( ( log n · r / R ) ) . By combining Theorems 3.1 and 3.2, we obtain the following.
Corollary 3.3.
Let L be a random n-dimensional lattice. Then with probability tending to as n → ∞ , itholds that for any n / -dimensional lattice subspace W of L and any r ≥ c · n / , | L ∩ W ∩ B n ( r ) | ≤ ( / ) exp ( ( log n · r / ( c n / )) ) . We will need the following point-counting version of Minkowski’s first theorem due to Blichfeldt andvan der Corput (see [4, Thm. 1 of Ch. 2, Sec. 7]).
Lemma 4.1 ([10]) . For any lattice L ⊂ R n with det ( L ) ≤ and r > , | L ∩ B n ( r ) | ≥ − n · vol ( B n ( r )) = √ π n (cid:16) π er n (cid:17) n / ( + o ( )) . Proof of Theorem 1.4.
By Lemma 4.1 applied to A = L ∩ B n ( n / ) , if n is large enough then | A | ≥ n n / . (2)Let G be a d -dimensional GAP as in Eq. (1) where d ≤ cn . Assume without loss of generality that theindices i are sorted in non-increasing order of b i − a i . For any t = ( t n / , . . . , t d ) where a i ≤ t i ≤ b i , considerthe restriction of the GAP obtained by fixing all but its first n / − G t = (cid:110) x + n / − ∑ i = α i x i + d ∑ i = n / t i x i : a i ≤ α i ≤ b i , α i ∈ Z (cid:111) . D ISCRETE A NALYSIS , 2017:8, 6pp. 4 C
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Let W t be an n / G t . By Corollary 3.3, | A ∩ G t | ≤ | L ∩ W t ∩ B n ( n / ) | ≤ ( / ) exp ( ( log n / c ) n / ) ≤ n , where the last inequality assumes that n is large enough. Therefore, | A ∩ G | ≤ ∑ t | A ∩ G t |≤ n · d ∏ i = n / ( b i − a i + ) ≤ n · | G | − ( n / − ) / d ≤ n · | A | − ( n / − ) / d ≤ ( n / | A | n / ( d ) ) · | A | . To conclude the proof note that | A | n / ( d ) ≥ | A | / ( c ) ≥ n n / ( c ) by Eq. (2). Therefore | A ∩ G | / | A | ≤ n − n / ( c ) assuming n is large enough. Acknowledgment
We thank Daniel Dadush for useful discussions.
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Shachar LovettComputer Science DepartmentUC San Diegoslovett ucsd edu http://cseweb.ucsd.edu/~slovett/home.html