Khintchine-type theorems for values of subhomogeneous functions at integer points

On common values of lacunary polynomials at integer points

For fixed ` ≥ 2, fixed positive integers m1 > m2 with gcd(m1,m2) = 1 and n1 > n2 > · · · > n` with gcd(n1, . . . , n`) = 1, and fixed rationals a1, a2, . . . , a`+1, b1, b2 which are all nonzero except for possibly a`+1, we show the finiteness of integral solutions x, y of the equation a1x n1 + · · ·+ a`x` + a`+1 = b1y + b2y , when n1 ≥ 3, m1 ≥ 2`(` − 1), and (n1, n2) 6= (m1,m2). In relation to...

Khintchine-type Theorems on Manifolds: the Convergence Case for Standard

Notation. The main objects of this paper are n-tuples y = (y1, . . . , yn) of real numbers viewed as linear forms, i.e. as row vectors. In what follows, y will always mean a row vector, and we will be interested in values of a linear form given by y at integer points q = (q1, . . . , qn) T , the latter being a column vector. Thus yq will stand for y1q1 + · · ·+ ynqn. Hopefully it will cause no ...

Rings of functions with integer derivatives at x=1

Special values of Green functions at big CM points

We give a formula for the values of automorphic Green functions on the special rational 0-cycles (big CM points) attached to certain maximal tori in the Shimura varieties associated to rational quadratic spaces of signature (2d, 2). Our approach depends on the fact that the Green functions in question are constructed as regularized theta lifts of harmonic weak Maass forms, and it involves the S...

Algebraic Values of Transcendental Functions at Algebraic Points

In this paper, the authors will prove that any subset of Q can be the exceptional set of some transcendental entire function. Furthermore, we could generate this theorem to a much more general version and present a unified proof.