9,323 research outputs found

### On odd covering systems with distinct moduli

A famous unsolved conjecture of P. Erdos and J. L. Selfridge states that
there does not exist a covering system {a_s(mod n_s)}_{s=1}^k with the moduli
n_1,...,n_k odd, distinct and greater than one. In this paper we show that if
such a covering system {a_s(mod n_s)}_{s=1}^k exists with n_1,...,n_k all
square-free, then the least common multiple of n_1,...,n_k has at least 22
prime divisors.Comment: 7 pages, final versio

### A variant of Tao's method with application to restricted sumsets

In this paper, we develop Terence Tao's harmonic analysis method and apply it
to restricted sumsets. The well known Cauchy-Davenport theorem asserts that if
$A$ and $B$ are nonempty subsets of $Z/pZ$ with $p$ a prime, then $|A+B|\ge
min{p,|A|+|B|-1}$, where $A+B={a+b: a\in A, b\in B}$. In 2005, Terence Tao gave
a harmonic analysis proof of the Cauchy-Davenport theorem, by applying a new
form of the uncertainty principle on Fourier transform. We modify Tao's method
so that it can be used to prove the following extension of the Erdos-Heilbronn
conjecture: If $A,B,S$ are nonempty subsets of $Z/pZ$ with $p$ a prime, then
$|{a+b: a\in A, b\in B, a-b not\in S}|\ge min {p,|A|+|B|-2|S|-1}$

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