2025/04/13 by Stephan Baier, Habibur Rahaman, Baier, Stephan +1
Mathematics · #Analytic Number Theory Research #Algebraic Geometry and Number Theory #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2504.09650
For any given positive definite binary quadratic form Q with integer coefficients, we establish two results on Diophantine approximation with integers represented by Q. Firstly, we show that for every irrational number α, there exist infinitely many positive integers n represented by Q and satisfying ||αn||0. This is an easy consequence of a result by Cook on small fractional parts of diagonal quadratic forms. Secondly, we give a quantitative version with a lower bound of this result when the exponent 1/2-ε is replaced by any fixed γ<3/7. To this end, we use the Voronoi summation formula and a bound for bilinear forms with Kloosterman sums to fixed moduli by Kerr, Shparlinski, Wu and Xi.