Abstract

Problems related to arithmetic properties of linear forms in various irrational and transcendental numbers, lower bounds for these linear forms, and approximation by rational fractions with bounded denominators are very popular in transcendence theory. We consider a new symmetric integral construction and prove a series of new lower bounds for joint approximations of certain values of the functions $\ln <!--FUNCTION\; APPLICATION-->x,\arctan <!--FUNCTION\; APPLICATION-->x$. Namely, the integral construction of such a type allows us to obtain new bounds for the linear forms $|p+q\cdot \ln <!--FUNCTION\; APPLICATION-->W+r\cdot \arctan <!--FUNCTION\; APPLICATION-->V|$ with special values of $W$, $V$.

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