H. Davenport

Rational approximations to algebraic numbers

It was proved in a recent paper that if α is any algebraic number, not rational, then for any ζ > 0 the inequality has only a finite number of solutions in relatively prime integers h, q . Our main purpose in the present note is to deduce, from the results of that paper, an explicit estimate for the number of solutions.

Note on irregularities of distribution

Let ( x 1 , y 1 ), …, ( x N , y N ) be N points in the square 0 ≤ x y S (ξ, η) denote the number of points of the set satisfying

Small Differences Between Prime Numbers

Let pn denote the nth prime number. The present investigation relates to the existence of relatively small values of pn+1─ pn when n is large, and establishes more precise results than those previously known.

Cubic forms in sixteen variables

It is proved that if C(x1, ..., xn) is any cubic form in n variables, with integral coefficients, then the equation C(x1, ..., xn) = 0 has a solution in integers x1, ..., xn not all 0, provided n is at least 16. This is an improvement upon earlier results (Davenport 1959, 1962).

Homogeneous Additive Equations

An investigation into conditions under which an equation of the form c1xk1 + ... + csxks = 0 will have solutions in which x1 ..., xs are integers (not all 0).

Simultaneous equations of additive type

The paper contains an investigation of conditions under which R simultaneous equations of additive type in N unknowns have a solution in integers, not all 0. If the degree k of the equations is odd, it suffices if N is greater than an explicit function of R and k. If k is even, two further conditions are imposed, and neither can be entirely avoided. It is also proved that the equations have a solution in p-adic integers, not all 0, if N is greater than an explicit function of R and k.

Cubic Forms in Thirty-Two Variables

It is proved that if C(xu...,*„) is any cubic form in n variables, with integral coefficients, then the equation C{xu ...,*„) = 0 has a solution in integers xXi...,xn, not all 0, provided n is at least 32. The proof is based on the Hardy-Littlewood method, involving the dissection into parts of a definite integral, but new principles are needed for estimating an exponential sum containing a general cubic form. The estimates obtained here are conditional on the form not splitting in a particular manner; when it does so split, the same treatment is applied to the new form, and ultimately the proof is made to depend on known results.

Simultaneous Diophantine approximation

The simplest problems of Diophantine approximation relate to the approximation of a single irrational number 0 by rational numbers pjq, and the principal question is how small we can make the error 0 — pjq in relation to q for infinitely many approximations. It is well known that this question can be answered almost completely in terms of the continued fraction expansion of 0. It must be admitted that our knowledge of the relationship between the continued fraction expansion of 0 and other possible representations of 0 is very fragmentary, a striking enough example being the number tf/2. However, the continued fraction theory gives us many general results which are best possible. Thus every 0 admits an infinity of approximations satisfying

Indefinite quadratic forms in many variables

Let Q ( x 1 , …, x n ) be an indefinite quadratic form in n variables with real coefficients. It is conjectured that, provided n ≥ 5, the inequality is soluble for every e > 0 in integers x 1 , …, x n , not all 0. The first progress towards proving this conjecture was made by Davenport in two recent papers; the result obtained involved, however, a condition on the type of the form as well as on n . We say that a non-singular Q is of type ( r, n—r ) if, when Q is expressed as a sum of squares of n real linear forms with positive and negative signs, there are r positive signs and n—r negative signs. It was proved that (1) is always soluble provided that

Cubic forms in 29 variables

It is proved that if C(x1 ..., xn) is any cubic form in n variables, with integral coefficients, then the equation C(x1 xn) = 0 has a solution in integers x1 ..., xn not all 0, provided n is at least 29. This is an improvement on a previous result (Davenport 1959).