33 research outputs found
On the Square Peg Problem and some Relatives
The Square Peg Problem asks whether every continuous simple closed planar
curve contains the four vertices of a square. This paper proves this for the
largest so far known class of curves.
Furthermore we solve an analogous Triangular Peg Problem affirmatively, state
topological intuition why the Rectangular Peg Problem should hold true, and
give a fruitful existence lemma of edge-regular polygons on curves. Finally, we
show that the problem of finding a regular octahedron on embedded spheres in
R^3 has a "topological counter-example", that is, a certain test map with
boundary condition exists.Comment: 15 pages, 14 figure
Optimal bounds for the colored Tverberg problem
We prove a "Tverberg type" multiple intersection theorem. It strengthens the
prime case of the original Tverberg theorem from 1966, as well as the
topological Tverberg theorem of Barany et al. (1980), by adding color
constraints. It also provides an improved bound for the (topological) colored
Tverberg problem of Barany & Larman (1992) that is tight in the prime case and
asymptotically optimal in the general case. The proof is based on relative
equivariant obstruction theory.Comment: 17 pages, 3 figures; revised version (February 2013), to appear in J.
European Math. Soc. (JEMS
Quadratic Fields Admitting Elliptic Curves with Rational -Invariant and Good Reduction Everywhere
Clemm and Trebat-Leder (2014) proved that the number of quadratic number
fields with absolute discriminant bounded by over which there exist
elliptic curves with good reduction everywhere and rational -invariant is
. In this paper, we assume the -conjecture to show
the sharp asymptotic for this number, obtaining
formulae for in both the real and imaginary cases. Our method has three
ingredients:
(1) We make progress towards a conjecture of Granville: Given a fixed
elliptic curve with short Weierstrass equation for
reducible , we show that the number of integers , , for which the quadratic twist has an integral
non--torsion point is at most , assuming the -conjecture.
(2) We apply the Selberg--Delange method to obtain a Tauberian theorem which
allows us to count integers satisfying certain congruences while also being
divisible only by certain primes.
(3) We show that for a polynomially sparse subset of the natural numbers, the
number of pairs of elements with least common multiple at most is
for some . We also exhibit a matching lower
bound.
If instead of the -conjecture we assume a particular tail bound, we can
prove all the aforementioned results and that the coefficient above is
greater in the real quadratic case than in the imaginary quadratic case, in
agreement with an experimentally observed bias.Comment: 35 pages, 1 figur