Sumsets avoiding squarefree integers

On primitive sets of squarefree integers

Consecutive Integers in High-multiplicity Sumsets

Sharpening (a particular case of) a result of Szemerédi and Vu [SV06] and extending earlier results of Sárközy [S89] and ourselves [L97b], we find, subject to some technical restrictions, a sharp threshold for the number of integer sets needed for their sumset to contain a block of consecutive integers of length, comparable with the lengths of the set summands. A corollary of our main result is...

Squarefree Integers without Large Prime Factors in Short Intervals

On Integers for Which the Sum of Divisors is the Square of the Squarefree Core

We study integers n > 1 satisfying the relation σ(n) = γ(n)2, where σ(n) and γ(n) are the sum of divisors and the product of distinct primes dividing n, respectively. We

Sequences of Integers Avoiding 3-term Arithmetic Progressions

The optimal length r(n) of a sequence in [1, n] containing no 3term arithmetic progression is determined for several new values of n and some results relating to the subadditivity of r are obtained. We also prove a particular case of a conjecture of Szekeres. A subsequence S = (a1, a2, . . . , ak) of the sequence 〈n〉 = (1, 2, . . . , n) containing no three terms ap, aq, and ar for which aq − ap...