#### IMSc Webinar

#### Diophantine Approximation with Prime Restriction in Real Quadratic Number Fields

#### Dwaipayan Mazumder

##### RKMVERI

*The distribution of $\alpha p$ modulo one, where $p$ runs over the rational primes and $\alpha$ is a fixed irrational real, has received a lot of attention. It is natural to ask for which exponents $*

u>0$ one can establish the infinitude of primes $p$ satisfying $||\alpha p||\le p^{-

u}$. The latest record in this regard is Kaisa Matom\"aki's landmark result $

u=1/3-\varepsilon$ which presents the limit of currently known technology. Prof. Marc Technau and Prof. Stephan Baier produced an analogue of Bob Vaughan's result $

u=1/4-\varepsilon$ for all imaginary quadratic number fields of class number 1. Prof. Stephan Baier and I proved an analog of the last-mentioned result for real quadratic fields of class number 1 under a certain Diophantine restriction.In the present article, we see an improvement of the relevant exponent to 7/22 for real quadratic fields of class number 1 under same Diophantine restriction.

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