Antal Balog

ON STRINGS OF CONSECUTIVE INTEGERS WITH NO LARGE PRIME FACTORS

We investigate conditions which ensure that systems of binomial polynomials with integer coefficients are simultaneously free of large prime factors. In particular, for each positive number e, we show that there are infinitely many strings of consecutive integers of size about n , free of prime factors exceeding n e , with the length of the strings tending to infinity with speed log log log log n .

New Sum-Product Estimates for Real and Complex Numbers

A variation on the sum-product problem seeks to show that a set which is defined by additive and multiplicative operations will always be large. In this paper, we prove new results of this type. In particular, we show that for any finite set

SIX PRIMES AND AN ALMOST PRIME IN FOUR LINEAR EQUATIONS

A

Multiplicative functions in arithmetic progressions

A of positive real numbers, it is true that

A low-energy decomposition theorem

\begin{aligned} \Big |\Big \{\frac{a+b}{c+d}:a,b,c,d\in {A}\Big \}\Big |\ge {2|A|^2-1}. \end{aligned}

Gaps between fractional parts, and additive combinatorics

|{a+bc+d:a,b,c,d∈A}|≥2|A|2-1.As a consequence of this result, it is also established that