Diophantine approximation with restricted denominators

Rahaman, Habibur (2026) Diophantine approximation with restricted denominators. PhD thesis, Indian Institute of Science Education and Research Kolkata.

Text (PhD thsis of Habibur Rahaman (21RS044)) 21RS044.pdf - Submitted Version Restricted to Repository staff only Download (2MB)

Abstract

The classical Dirichlet approximation theorem implies that, for any irrational number ɑ, there are infinitely many pairs of coprime integers a, q with q positive, satisfying |ɑq - a| < 1/q. In other words, the distance of ωq from the nearest integer |ɑq|, satisfies |ɑq| < q⁻¹, for infinitely many positive integers q. The aim of this thesis is to study this problem, where the denominators q are restricted to certain sparse subsets of the set of positive integers and in each case to obtain the best exponents of q currently achievable in place of -1 improving upon earlier results. In Chapter 3, we consider the problem where q’s are restricted to the set of Pyateckiǐ-Šapiro primes, that is, primes of the form [nc], for some 1 < c < 2. It is known that there are infinitely many such primes and in fact, they form a power saving sparse subset of the set of primes. Dimitrov has shown the existence of infinitely many such primes p satisfying |ɑp + β| < p⁻¹²⁻¹¹c/²⁶c , for any irrational number ɑ, any real number β, and for 1 < c < 12/11. Using Harman’s sieve with exponential sums, we improve this result in two directions, obtaining a better exponent and widening the range of c to 1 < c < 9/8. Let Q(x, y) be any positive definite binary quadratic form with integer coe!- cients. Due to Landau, it is known that the set of values of Q at integer inputs forms a sparse subset of the set of positive integers. In Chapter 4, we consider the Diophantine approximation problem, where q’s are restricted to the set of integer values taken by Q(x, y) and prove the existence of infinitely many such q with an exponent -1/2, improving previous results. We then obtain a quantitative result for such q in intervals, which is very close to the expected number of such q, although with a weaker exponent -3/7 instead of -1/2. In Chapter 5, we further investigate this problem and obtain the same quantitative result with exponent →1/2, but for the particular case where Q(x, y) = x² + y². In Chapter 6, we consider the Diophantine approximation problem, where the denominators are running over y-smooth numbers, i.e., numbers which are free of prime factors larger than y. For su!ciently smaller values of the smoothness parameter y, the set of y-smooth numbers forms a power saving sparse subset of the set of positive integers.

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