On the probability that n and [n c] are coprime

We prove that for any positive real number which is not an integer, the density of the integers which are coprime to , a result conjectured by Moser, Lambek and Erd Hs. This revised version was published online in June 2006 with corrections to the Cover Date.     

7373170279850

We conjecture that 7,373,170,279,850 is the largest integer which cannot be expressed as the sum of four nonnegative integral cubes.

     

Distribution modulo 1 of a linear sequence associated to a multiplicative function evaluated at polynomial arguments

In two previous papers,the first named author jointly with Florian Luca and Henryk Iwaniec,have studied the distribution modulo 1 of sequences which have linear growth and are mean values of multiplicative functions on the set of all the integers.In this note,we give a first result concerning sequences with linear growth associated to the mean values of multiplicative functions on a set of polynomial values,proving the density modulo 1 of the sequencem[∑((m2+1))(m2+1)(m≤n)]n.This result is but an illustration of the theme which is currently being developed in the PhD thesis of the second named author.     

A refined bound for sum-free sets in groups of prime order

Improving upon earlier results of Freiman and the present authors,we show that if p is a sufficiently large prime and A is a sum-freesubset of the group of order p, such that n: = |A| > 0.318p,then A is contained in a dilation of the interval [n, p –n](mod p).     

When subset-sums do not cover all the residues modulo p

Let . We prove that a subset of , where p is a prime number, with cardinality larger than such that its subset sums do not cover has an automorphic image which is rather concentrated; more precisely, there exists s prime to p such that

     

On a question about sum-free sequences

An increasing sequence of positive integers {n₁, n₂, …} is called a sum-free sequence if every term is never a sum of distinct smaller terms. We prove that there exist sum-free sequences {n_k} with polynomial growth and such that lim_k→∞ n_k+1/n_k = 1.     

On an additive problem of Erdős and Straus, 1

According to Erdős and Straus, we define an admissible subsetA of [1,N] to be such that whenever an integer can be written as a sum ofs distinct elements fromA, thens is well defined. Improving on previous results, we show that the cardinality of such an admissible subsetA is at most (2 +o(1))√N. As shown by Straus, the constant 2 cannot be improved upon.     

On Bounds for the Concentration Function II

We derive an upper bound for the concentration of the sum of i.i.d. random variables with values in by appealing to functions of positive type and the structure theory of set addition.