Littlewood’s Conjecture For The L~1Norm Of Exponential Sums

Let S be a finite set of integers and Fs(x)=∑2πiαθ be its exponential sum. McGe-hee,Pigno,Smith and Konyagin have independently proved that tFs(x)≥clog|S|for some absolute constant c. The lower bound has the correct order of magnitude and was first conjectured by Littlewood. In this paper,we present a lower bounds on the Ll-norm of exponential sums.Through analysizing the properties of the function, n=1and n=2, we can get specific integral value.Thus we can identify an absolute positive constent C so that inequality was established.By using the properties of complex number:A plural mode is no less than the absolute value of the plural real component.We can construct a function,We will be deformation of inequality.By constructing the inner product and norm,we can use Fourier series’knowlege to manage the inner product and norm,so we can get a new inequality.the original problem can be converted into seeking exponetial sums of the function’s absolute value.In this paper, the key step is to construct a function.By using the related knowlege of the nonpositive support and functional approximation,we can get a lower bound of the funtion. After calculation, we can work out a absolute positive constant,so the theorem of Littlewood’s conjecture is proved.