Applications of the Subspace Theorem to Certain Diophantine Problems

Perfect powers with few binary digits and related Diophantine problems, II

“…As noted by Corvaja and Zannier (see page 169 of [11]), the rather curious problems we consider here fit into a more general framework, as solutions to such polynomial-exponential equations correspond to S-integral points on certain projective varieties, for suitable sets of primes S. Viewed in this light, finiteness statements for these equations would follow from essentially the simplest open case of a deep conjecture of Lang-Vojta on Zariski denseness of S-integral points on varieties of log-general type (see e.g. page 486 of [13]).…”

Combinatorial applications of the subspace theorem

“…Clearly b ≥ n/8 and a + b ≤ n since AOB is a substring of b n . Since α starts with b n we have 8 . Now we apply the Subspace Theorem.…”

“…A number of surveys have been given of the Subspace Theorem highlighting its multitude of applications. Notable surveys include those of Bilu [4], Evertse and Schlickewei [16] and Corvaja and Zannier [8]. These give many proofs of results from number theory and algebraic geometry using the Subspace Theorem including those mentioned above.…”

On sparse perfect powers

“…This class of problems also presents some ties to algebraic geometry. In fact, Corvaja and Zannier showed in [3] that solutions of an equation of the form (1) are associated with Sintegral points on certain projective varieties. For instance, assume for the sake of simplicity that x = p is a prime number, and that k, d are fixed and c 0 = c 1 = • • • = c k−1 = 1 in equation (1).…”