We use Green's transference principle to show that any subset of the dth powers of primes with positive relative density contains nontrivial solutions to a translation-invariant linear equation in d 2 + 1 or more variables, with explicit quantitative bounds.

Abstract. Under fairly natural assumptions, Huang counted the number of rational points lying close to an arc of a planar curve. He obtained upper and lower bounds of the correct order of magnitude, and conjectured an asymptotic formula. In this note, we establish the conjectured asymptotic formula.

The sequence 3, 5, 9, 11, 15, 19, 21, 25, 29, 35, . . . consists of odd legs in right triangles with integer side lengths and prime hypotenuse. We show that the upper density of this sequence is zero, with logarithmic decay. The same estimate holds for the sequence of even legs in such triangles. We expect our upper bound, which involves the Erdős-Ford-Tenenbaum constant, to be sharp up to a double-logarithmic factor. We also provide a nontrivial lower bound. Our techniques involve sieve methods, the distribution of Gaussian primes in narrow sectors, and the Hardy-Ramanujan inequality.

A famous result due to Birch (1961) provides an asymptotic formula for the number of integer points in an expanding box at which given rational forms of the same degree simultaneously vanish, subject to a geometric condition. We present the first inequalities analogue of Birch's theorem.2010 Mathematics Subject Classification. 11D75, 11E76. Key words and phrases. Diophantine inequalities, forms in many variables, inhomogeneous polynomials.