科学文献

1 Research
2
V. BENTKUS
F. GOTZE
1. Introduction and results
Let R d , 1 d < 1, denote a real d-dimensional Euclidean space with scalar product ; and norm
jxj = x; x = x +
2 2 1
+ x2 ; d
for x = (x1; : : :; xd ) 2 R d :
Let Zd be the standard lattice of points with integer coordinates in R d . For a (measurable) set B R d let vol B denote the Lebesgue measure of B , and let volZ B denote the lattice volume of B , that is the number of points in B \ Zd. For p 2 N , p 2, consider a polynomial
LATTICE POINTS IN MULTIDIMENSIONAL BODIES
V. Bentkus1 F. Gotze1
University of Bielefeld November 1998
Abstract. Asymptotic expansions for the number of lattice points in bounded regions of
q jxjp
with some q > 0. De ne the region
Qp(
x)
kQ p k jxjp
(1.5)
Es = x 2 R d : Q (x) s ;
for s 2 R :
(1.6)
For positive de nite Q p the region Es is a compact subset of R d , and Es 6= ?, for s 0. Hence, there exists a minimal number r = rs 1 such that the cube
LATTICE POINTS
3
in order to avoid additional technicalities associated with the inde nite case, cf. proofs in both cases for quadratic forms in Bentkus and Gotze (1997c) (henceforth brie y BG). For positive de nite Q p we have
^ j d0 j; =j j_ where a _ b = maxfa; bg and a ^ b = minfa; bg. Let
1
= j 1j ^
_ j d0 j;
(1.2) (1.3)
B p(
x; d0) = B p x; : : :; x] =
Qp(
d0 P j =1
p j xj ;
x = (x1 ; : : :; xd ) 2 R d ;
B (r) = x 2 R d : jxj1 r ; jxj1 = jx1j _ contains the set Es. Clearly, rs = 1 _ sup jxj1 :
Write as well
x2Es
_ jxd j;
(1.7)
r = sup jxj1 :
x2E1
(1.8)
For Q with a positive de nite leading part the number r satis es 0 < r < 1. Furthermore, for such Q we have lims!1 rs =s1=p = r , which means that asymptotically the quantity rs is proportional to s1=p , as s ! 1 (see Proposition 3.5). There are two natural interrelated lattice points problems associated to a a polynomial. The rst one is to investigate the asymptotic as s ! 1 of volZ Es = volZ x 2 Zd : Q (x) s ; (1.9) for given polynomial Q . Another way is to blow up a body, that is, to study the asymptotic of volZ(r E1) = volZ r x 2 Zd : Q (x) 1 (1.10) as r ! 1. For both bodies (1.9) and (1.10), we shall obtain bounds for the lattice point remainders (s; Q ) def jvolZ Es ? vol Esj and = It is clear that (r; Q ) def jvolZ rE1 ? vol rE1j: = + rp?1 Q 1 (x) rp : (1.11) (1.12)
Q(
x) =
p P j =1
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Qj (
x);
deg Q j = j;
x 2 Rd ;
jxj
1
(1.1)
where Q j denotes the j -th homogeneous part of Q . We shall write kQj k = sup Qj (x) . For j 6= 0, j 2 R , and 1 d0 d, write
d0 P j =1
xp + j
d0 P j =1
x2 j
d P j =d0 +1
p xj ?2 ;
p > 2;
has the form (1.4). We call a polynomial Q p positive de nite if Q p (x) > 0, for all x 6= 0. Otherwise Q p is called inde nite. In this Introduction we shall restrict ourselves to problems related to positive de nite polynomials Q with the leading part of the form (1.4). The reason is that in order to apply our main result|Theorem 2.2, we have to verify a condition (see (2.8)) for trigonometric sums. This is done in Section 8 for polynomials with the leading part (1.4). In more general situations one has to develop a technique more re ned than that used in Section 8. We assume positive de niteness in this Introduction only
and
x) = B p (x; d0) + Pp (x); (1.4) where Pp : R d ! R , deg Pp = p, is a homogeneous polynomial such that its degree in the variables x1; : : : ; xd0 is strictly smaller than p. For example, the polynomial
the d-dimensional Euclidean space are obtained. We assume that the regions are level sets of polynomials of degree p 2. The expansions are valid provided that some 'local' bounds on trigonometric sums related to these polynomials are satis ed. Using this approach we obtain as an initial application optimal bounds for the lattice point rest in large bodies generated by polynomials such as Q(x) = Qp (x) + P(x), deg P < p, x 2 Rd , where Qp has the form Qp (x) = 1 xp + + d xp with some j > 0, for 1 j d, provided that the 1 d dimension d cp is su ciently large. The proofs are based on an extension of techniques developed for quadratic forms by the authors. Similar to the quadratic case the results allow to prove variants of the quantitative Oppenheim conjecture and the Davenport and Lewis conjecture for classes of higher order polynomials assuming that the dimension is su ciently large.
Contents
1. Introduction and results.......................................................................................2 2. The main result....................................................................................................9 3. An application of the main result: proofs of theorems of the introduction.............19 b 4. An expansion for the Fourier{Stieltjes transform F ..............................................26 5. Multiplicative inequalities for trigonometric sums................................................35 6. The integration procedure for large jtj..................................................................44 7. Symmetrization inequalities................................................................................45 8. Upper bounds for trigonometric sums..................................................................49 9. Trigonometric sums of irrational polynomials......................................................57 10. References..........................................................................................................61
supported by the SFB 343 in Bielefeld. 1991 Mathematics Subject Classi cation. 11P21. Key words and phrases. lattice points, lattice rest for multidimensional regions, distribution of values of polynomials, Oppenheim's conjecture, Davenport and Lewis conjecture. Typeset by AMS-TEX 1