##### Department of Mathematics,

University of California San Diego

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### Math 269 - Combinatorics

## Vsevolod (Seva) Lev

#### The University of Haifa \\ University of California, San Diego

## Projecting difference sets onto the positive orthant

##### Abstract:

A combinatorial geometry problem, related (in a surprising way) to the Graham's g.c.d. conjecture, is as follows. Let $n\ge 1$ be an integer. Given a vector $(a_1 , ... , a_n)\in R^n$, write $$ a^+ := ( \max(a_1,0) , ... , \max(a_n,0) ) $$ (the "projection of $a$ onto the positive orthant"), and for a set $A\subset R^n$ put $$ A^+ := \{ a^+ : a\in A \}. $$ How small $|(A-A)^+|$ can be for a set $A\subset R^n$ of given cardinality $|A|$? We discuss previously known results and report on recent developments due to Ron Holzman, Rom Pinchasi, and the presenter.

### November 20, 2007

### 3:00 PM

### AP&M 7321

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