A question related to sum-product

Additive Combinatorics

SumProduct

Above is a pdf file about an question related to the Sum-product phenomenon introduced by Erdos and Szemeredi.

Fix A \subset \mathbb{R}^+ to be finite and nonempty. We are interested in nontrivial lower bounds for |A + A \cdot A|. The best known bounds in such generality come from the Szemeredi-Trotter theorem from incidence geometry giving |A + A \cdot A| \gg |A|^{3/2}. Though when one restricts to A \subset \mathbb{Z}^+, one has |A+A \cdot A| \geq |A|^2 + |A| -1. We give a short proof of this fact in the above file.

Recently, Oliver Roche-Newton, Imre Ruzsa, Chen-Yen Shen and Ilya Shkredov have made progress! They show both that the previously known upper and lower bounds are not sharp. It is still certainly plausible that for any $\epsilon > 0$, |AA+A| \gg_{\epsilon} |A|^{2- \epsilon}.

Closed Unit Balls and Quotient Maps

Functional Analysis

ClosedUnitBall

Above is a pdf about closed unit balls and quotient maps. This was the result of several discussions with Chris Gartland, a fellow graduate student.

Fix a Banach space V and a closed subspace M. Elementary considerations assert that the open unit ball of V is mapped onto the open unit ball V / M equipped with the usual norm (this should be compared to a general transformation where even in finite dimensions the image is an ellipse).  What happens to the closed unit ball V? It is easy to see that the projection map is a contraction, so its image is contained in the closed unit ball of V / M. Is the map onto in general? If not, what if we put some additional restrictions on V? In the above pdf we investigate this question which illustrates the usefulness of equipping V with different topologies.