ar X iv : m at h / 04 07 50 3 v 1 [ m at h . M G ] 2 8 Ju l 2 00 4 SEPARATED SETS AND THE FALCONER CONJECTURE FOR POLYGONAL NORMS

The Falconer conjecture [F86] asserts that if E is a planar set with Hausdorff dimension strictly greater than 1, then its Euclidean distance set ∆(E) has positive one-dimensional Lebesgue measure. We discuss the analogous question with the Euclidean distance replaced by non-Euclidean norms ‖ · ‖X in which the unit ball is a polygon with 2K sides. We prove that for any such norm, and for any α > K/(K − 1), there is a set of Hausdorff dimension α whose distance set has Lebesgue measure 0. Mathematics Subject Classification: 28A78. §0. INTRODUCTION A conjecture of Falconer [F86] asserts that if a set E ⊂ R has Hausdorff dimension strictly greater than 1, then its Euclidean distance set ∆(E) = ∆l2 2 (E) = { ‖x− x‖l2 2 : x, x ∈ E } has positive one-dimensional Lebesgue measure. The current best result in this direction is due to Wolff [W99], who proved that the conclusion is true if E has Hausdorff dimension greater than 4/3. Erdogan [Er03] extended this result to higher dimensions, proving that the same conclusion holds for subsets of R with Hausdorff dimension greater than d(d + 2)/2(d+ 1). This improves on the earlier results of Falconer [F86], Mattila [M87], and Bourgain [B94]. A similar question can be posed for more general two-dimensional normed spaces. More precisely, if X is such a space and E ⊂ X , then we define the X-distance set of A as ∆X(E) = {‖x− x‖X : x, x ∈ E} and ask how the size of ∆X(E) depends on the dimension of E as well as on the properties of the norm ‖ · ‖X . Simple examples show that Falconer’s conjecture as stated above, but with ∆(E) replaced by ∆X(E), cannot hold for all normed spaces X . For instance, let ‖x‖l2 ∞ = max(|x1|, |x2|) and let E = F × F , where F is a subset of [0, 1] with Hausdorff dimension 1 such that F − F := {x − x : x, x ∈ F} has measure 0. (It is an easy exercise to modify the Cantor set construction to produce such a set.) Then E has Hausdorff dimension 2, but its l ∞-distance set F − F has measure 0. Here and below, we use dim(E) to denote the Hausdorff dimension of E, |F |d to denote the d-dimensional Lebesgue measure of F , and |A| to denote the cardinality of a finite set A. Typeset by AMS-TEX 1 Definition 0.1.. Let 0 < α < 2. We will say that the α-Falconer conjecture holds in X if for any set E ⊂ X with dim(E) > α we have |∆X(E)|1 > 0. Iosevich and the second author [I L04] proved that the 3/2-Falconer conjecture holds if the unit ball in X , BX = {x ∈ R : ‖x‖X ≤ 1}, is strictly convex and its boundary ∂BX has everywhere nonvanishing curvature, in the sense that the diameter of the chord {x ∈ BX : x · v ≥ max y∈BX (y · v)− ǫ}, where v is a unit vector and ǫ > 0, is bounded by C √ ǫ uniformly for all v and ǫ. We do not know of any counterexamples to the 1-Falconer conjecture in normed spaces with BX strictly convex. On the other hand, if BX is a polygon, then the above example shows that the α-Falconer conjecture may fail for all α < 2. The purpose of this paper is to examine this situation in more detail. Theorem 1. Let BX be a symmetric convex polygon with 2K sides. Then there is a set E ⊂ [0, 1] with Hausdorff dimension ≥ K/(K − 1) such that |∆X(E)|1 = 0. If we assume that there is a coordinate system in which the slopes of all sides of K are algebraic, then a stronger result is known [K L04]. Corollary 2. [K L04] If BX is a polygon with finitely many sides, and if there is a coordinate system in which all sides of BX have algebraic slopes, then there is a compact E ⊂ X such that the Hausdorff dimension of E is 2 and the Lebesgue measure of ∆X(E) is 0. In particular, Corollary 2 can be applied to all polygons BX with 4 or 6 sides. We do not know if the same assertion is true for all polygonal norms. However, using recent results on Diophantine approximations, one can prove it for almost all polygons BX . Fixing a coordinate system, we can define, for any non-degenerate segment I ⊂ X , its slope Sl(I): if the line containing I is given by an equation u1x1 + u2x2 + u0 = 0, then we set Sl(I) = −u1/u2. We write Sl(I) = ∞ if u2 = 0. Theorem 3. For any integer K ≥ 2 and for almost all γ1, . . . , γK the following is true. If BX is a symmetric convex polygon with 2K sides, and the slopes of non-parallel sides are equal to γ1, . . . , γK , then there is a compact E ⊂ X such that the Hausdorff dimension of E is 2 and the Lebesgue measure of ∆X(E) is 0. Actually, we will prove the stronger result: if the slopes of 3 non-parallel sides of BX are fixed, then for almost all choices of slopes of other K − 3 non-parallel sides the required compact A exists (recall that for K ≤ 3 Theorem 3 follows from Corollary 2).