m at h . C A ] 9 M ay 2 00 3 FALCONER CONJECTURE IN THE PLANE FOR RANDOM METRICS
نویسندگان
چکیده
The Falconer conjecture says that if a compact planar set has Hausdorff dimension > 1, then the Euclidean distance set ∆(E) = {|x − y| : x, y ∈ E} has positive Lebesgue measure. In this paper we prove, under the same assumptions, that for almost every ellipse K, ∆ K (E) = {||x − y|| K : x, y ∈ E} has positive Lebesgue measure, where || · || K is the norm induced by an ellipse K. Equivalently, we prove that if a compact planar set has Hausdorff dimension > 1, then ∆(T E) has positive Lebesgue measure for almost every transformations T with bounded positive eigenvalues. We also use this result to deduce a version of the Erdos Distance Conjecture in the plane.
منابع مشابه
Circular Averages and Falconer/erdos Distance Conjecture in the Plane for Random Metrics
We study a variant of the Falconer distance problem for perturbations of the Euclidean and related metrics. We prove that Mattila’s criterion, expressed in terms of circular averages, which would imply the Falconer conjecture, holds on average. We also use a diophantine conversion mechanism to prove that well-distributed case of the Erdos Distance Conjecture holds for almost every appropriate p...
متن کاملar X iv : m at h / 04 11 14 0 v 2 [ m at h . N T ] 4 M ay 2 00 5 THE 3 x + 1 SEMIGROUP
The 3x + 1 semigroup is the multiplicative semigroup S of positive rational numbers generated by { 2k+1 3k+2 : k ≥ 0} together with {2}. This semigroup encodes backwards iteration under the 3x + 1 map, and the 3x + 1 conjecture implies that it contains every positive integer. This semigroup is proved to be the set of positive rationals a b in lowest terms with b 6≡ 0( mod 3), and so contains al...
متن کاملOn a conjecture of a bound for the exponent of the Schur multiplier of a finite $p$-group
Let $G$ be a $p$-group of nilpotency class $k$ with finite exponent $exp(G)$ and let $m=lfloorlog_pk floor$. We show that $exp(M^{(c)}(G))$ divides $exp(G)p^{m(k-1)}$, for all $cgeq1$, where $M^{(c)}(G)$ denotes the c-nilpotent multiplier of $G$. This implies that $exp( M(G))$ divides $exp(G)$, for all finite $p$-groups of class at most $p-1$. Moreover, we show that our result is an improvement...
متن کاملm at h . A G ] 1 6 M ay 2 00 3 Green ’ s canonical syzygy conjecture for generic curves of odd genus Claire Voisin
The direction ⇒ is proved by Green and Lazarsfeld in the appendix to [3]. The case p = g − 2 of the conjecture is equivalent to Noether’s theorem, and the case p = g − 3 to Petri’s theorem (see [5]). The case p = g − 4 has been proved in any genus by Schreyer [9] and by the author [12] for g > 10. More recently, the conjecture has been studied in [10], [11], for generic curves of fixed gonality...
متن کاملm at h . C O / 0 20 52 06 v 1 1 9 M ay 2 00 2 132 - avoiding Two - stack Sortable Permutations , Fibonacci Numbers , and Pell Numbers ∗
In [W2] West conjectured that there are 2(3n)!/((n+1)!(2n+1)!) two-stack sortable permutations on n letters. This conjecture was proved analytically by Zeilberger in [Z]. Later, Dulucq, Gire, and Guibert [DGG] gave a combinatorial proof of this conjecture. In the present paper we study generating functions for the number of two-stack sortable permutations on n letters avoiding (or containing ex...
متن کامل