Set-polynomials and polynomial extension of the Hales-Jewett theorem.
Sets of integers that do not contain long arithmetic progressions.
Sets of lengths do not characterize numerical monoids.
Skolem–Mahler–Lech type theorems and Picard–Vessiot theory
We show that three problems involving linear difference equations with rational function coefficients are essentially equivalent. The first problem is the generalization of the classical Skolem–Mahler–Lech theorem to rational function coefficients. The second problem is whether or not for a given linear difference equation there exists a Picard–Vessiot extension inside the ring of sequences. The third problem is a certain special case of the dynamical Mordell–Lang conjecture. This allows us to deduce...
Solving a ± b = 2c in elements of finite sets
We show that if A and B are finite sets of real numbers, then the number of triples (a,b,c) ∈ A × B × (A ∪ B) with a + b = 2c is at most (0.15+o(1))(|A|+|B|)² as |A| + |B| → ∞. As a corollary, if A is antisymmetric (that is, A ∩ (-A) = ∅), then there are at most (0.3+o(1))|A|² triples (a,b,c) with a,b,c ∈ A and a - b = 2c. In the general case where A is not necessarily antisymmetric, we show that the number of triples (a,b,c) with a,b,c ∈ A and a - b = 2c is at most (0.5+o(1))|A|². These estimates...
Some divisibility properties of binomial coefficients and the converse of Wolstenholme's theorem.
Some extensions and refinements of a theorem of Sylvester
Some new exact van der Waerden numbers.
Some properties of a certain nonaveraging sequence.
Squares in arithmetical progressions I.
Squares in arithmetical progressions II.
Squares in products with terms in an arithmetic progression
Subset sums avoiding quadratic nonresidues
Sumsets containing long arithmetic progressions and powers of 2
Sur la fonction plus grand facteur premier
Sur la somme des puissances semblables des termes d'une progression arithmétique