Ernie Croot

activity

19992005

most citedThe Minimal Number of Three-Term Arithmetic Progressions Modulo a Prime Converges to a Limit

2 citations · 5 across the 11 of their papers we have counts for

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math.NT2004

Complexity of Inverting the Euler Function

Scott Contini, Ernie Croot, Igor Shparlinski

We present an algorithm to invert the Euler function $ϕ (m)$ . The algorithm, for a given $n \geq 1$ , in polynomial time ``on average'', finds the set $Ψ (n)$ of all solutions $m$ to…

math.NT2004

Long Arithmetic Progressions in Critical Sets

Ernie Croot

In this paper we prove: If 0 < d < 1, and p is a sufficiently large prime, then if S is a subset of Z/pZ having the least number of three-term arithmetic progressions among all sub…

math.NT2004★ 1 cited

Sums of the Form 1/x_1^k + ... + 1/x_n^k Modulo a Prime

Ernie Croot

We show that for every $0 < ε \leq 1$ and integer $k \geq 1$ , there exists an integer $n = n (ε, k)$ so that for all primes $p$ , and integers $0 \leq a \leq p - 1$ , there exist integers…

math.NT2004

k-term Arithmetic Progressions in Sumsets

Ernie Croot

In this paper we give a very elementary proof that if A and B are subsets of {1,2,...,N}, each having at least 5N^{1 - (4(k-1))^{-1}} elements, then the sumset A+B has a k-term ari…

math.NT2003

A Combinatorial Method for Counting Smooth Numbers in Sets of Integers

Ernie Croot

In this paper we present a method for producing asymptotic estimates for the number of integers in a given S having only ``small'' prime factors. The conditions that need to be ver…

math.NT2003

On a coloring conjecture about unit fractions

Ernest S. Croot

We prove an old conjecture of Erd{\H o}s and Graham on sums of unit fractions: There exists a constant $b > 0$ such that if we $r$ -color the integers in $2, b^{r}]$ , then there exists a…