On explicit versions of Tartakovski's theorem

Abstract

We give an explicit bound for representation of numbers by quaternary integral positive definite quadratic forms. 0. Introduction. Let q(x 1 , . . . , x m ) = m i, j=1a ij x i x j be an integral positive definite quadratic form in m м 4 variables. A famous theorem of Tartakovski states that any sufficiently large positive integer a for which the congruence q(x 1 , . . . , x m ) ≡ a mod c is solvable for all c ∈ Z with integral coprime x 1 , . . . , x m is represented (primitively) by the form q. This can be proved using the Hardy-Littlewood circle method or using modular forms; an arithmetic proof has been given in [7], [6].It is well-known that the result is effective, i.e., that one can specify a constant C(q) depending on q such that all a м C(q) are sufficiently large in this sense. Such explicit versions have (for m м 5) been discussed by Watson [9] and by Hsia/Icaza ([6]) and for m = 4 in [2], [3].It is therefore in principle possible to decide in finitely many steps which numbers outside the finitely many arithmetic progressions that are excluded by the congruence conditions above are represented (primitively) by the form. Nevertheless it is in general not feasible to do this using a computer since the size of the constant C(q) is either unknown or too large for practical purposes.In this note we try therefore to get as good and as explicit as possible estimates on the size of the constant C(q) using the modular forms method in the case m = 4. The approach is essentially that of [2], [3], taking care to evaluate constants explicitly. Cusp form estimates.Let V be a vector space of dimension m over Q, equipped with the positive definite quadratic form q : V −→ Q with associated symmetric bilinear form B(x, y) = q(x + y) − q(x) − q(y), M ֤ V a lattice of rank m with q(M) ֤ Z. The p-adic completions M ⊗ Z Z p are also denoted by M p , with M ∞ = V ⊗ Q R = V ∞ .

Ergodic optimization for countable alphabet subshifts of finite type

“…An arithmetic proof of Theorem 2.2 can be found in [2,Section 11.9]. An estimate on the size of c * (L) has been worked out in [15].…”

On the representation of integers by quadratic forms

“…The corresponding result for n = 4 is due to Fomenko [10] and uses the theory of modular forms. Schulze-Pillot [17] has since refined the argument, obtaining…”

Quadratic Forms and Semiclassical Eigenfunction Hypothesis for Flat Tori

“…Similar uniform results in a different context (explicit bounds for representability by a quadratic form) has been extensively studied by various authors. In particular, there is PhD work of Hanke [Han04] who uses theta series to get estimates which are uniform in the coefficients and also the work of Schulze-Pillot [SP01]. More recently, Browning and Deitmann [BD08, Proposition 1] established a result that recovers our theorem for in the generic situation where the coefficients of the quadratic form is of order D 1/k .…”