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Submitted 1 March, 2017; originally announced March 2017.

Comments: 6 pages; comments welcome!

arXiv:1612.06153 [, , other ] math.CO

On Cartesian Products which Determine Few Distinct Distances

Abstract : Every set of points \mathcal{P} P determines Ω(|\mathcal{P}| / \log |\mathcal{P}|) distances. A close version of this was initially conjectured by Erdős in 1946 and rather recently proved by Guth and Katz. We show that when near this lower bound, a point set \mathcal{P} of the form A \times A must satisfy |A - A| \ll |A|^{2-\frac{2}{7}} \log^{\frac{1}{7}} |A| . This improves recent results… ▽ More Every set of points \mathcal{P} determines Ω(|\mathcal{P}| / \log |\mathcal{P}|) distances. A close version of this was initially conjectured by Erdős in 1946 and rather recently proved by Guth and Katz. We show that when near this lower bound, a point set \mathcal{P} of the form A \times A must satisfy |A - A| \ll |A|^{2-\frac{2}{7}} \log^{\frac{1}{7}} |A| . This improves recent results of Hanson and Roche-Newton. △ Less

Submitted 25 March, 2018; v1 submitted 19 December, 2016; originally announced December 2016.

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math.CO

Boole's formula as a consequence of Lagrange's Interpolating Polynomial theorem

Abstract : We present a slightly more general version of Boole's additive formula for factorials as a simple consequence of Lagrange's Interpolating Polynomial theorem. ▽ More We present a slightly more general version of Boole's additive formula for factorials as a simple consequence of Lagrange's Interpolating Polynomial theorem. △ Less

Submitted 7 July, 2008; originally announced July 2008.

Comments: 3 pages

MSC Class: 11-03; 01A60

Journal ref: Integers: Electronic Journal of Combinatorial Number Theory 8 (2008), A23

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[, other ] math.HO

Generalization of the Apollonius Circles

Authors: HUGS IDEA Fashion Flower Low Top Shoes Women Platform Sneakers Colorful 3 o488N
, Vladimir Zajic

Abstract : The three Apollonius circles of a triangle, each passing through a triangle vertex, the corresponding vertex of the cevian triangle of the incenter and the corresponding vertex of the circumcevian triangle of the symmedian point, are coaxal. Similarly defined three circles remain coaxal, when the circumcevian triangle is defined with respect to any point on the triangle circumconic through the i… ▽ More The three Apollonius circles of a triangle, each passing through a triangle vertex, the corresponding vertex of the cevian triangle of the incenter and the corresponding vertex of the circumcevian triangle of the symmedian point, are coaxal. Similarly defined three circles remain coaxal, when the circumcevian triangle is defined with respect to any point on the triangle circumconic through the incenter and symmedian point. Inversion in the incircle of the reference triangle carries these three coaxal circles into coaxal circles, each passing through a vertex of the inverted triangle and centered on the opposite sideline, at the intersection of the orthotransversal with respect to a point on the Euler line of the inverted triangle. A similar circumconic exists in a more general configuration, when the cevian triangle is defined with respect to an arbitrary point, passing through this arbitrary point and isogonal conjugate of its complement. △ Less

BILL JOHNSON
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